Probability on graphs

Geoffrey Grimmett

Probability on Graphs Lecture Notes on Stochastic Processes on Graphs and Lattices

c G. R. Grimmett 6 February 2009

Geoffrey Grimmett Statistical Laboratory Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom

2000 MSC: (Primary) 60K35, 82B20, (Secondary) 05C80, 82B43, 82C22 With 42 Figures

c G. R. Grimmett 6 February 2009

Preface

Within the menagerie of objects studied in contemporary probability theory, there are a number of related animals that have attracted great interest amongst probabilists and physicists in recent years. The overall target of these notes is to identify some of these, and to develop their basic theory. The inspiration for many of these objects comes from physics, but the mathematical subject has taken on a life of its own, and many beautiful constructions have emerged. If the two principal characters in these notes are random walk and percolation, they are only part of the rich theory of uniform spanning trees, self-avoiding walks, random networks, models for ferromagnetism and the spread of disease, and motion in random environments. This is an area that has attracted many fine scientists, by virtue, perhaps, of its special mixture of modelling and problem-solving. There remain many open problems. It is the experience of the author that these may be explained successfully to a graduate audience open to inspiration and provocation. The material described here may be used as the basis of lecture courses of between 24 and 48 hours duration. Little is assumed about the mathematical background of the audience beyond some basic probability theory, but listeners should be willing to get their hands dirty if they are to profit. Care should be taken in the setting of examinations, since problems can be unexpectedly difficult. Successful examinations may be designed, and some help is offered through the inclusion of exercises at the ends of chapters. As an alternative to a conventional examination, students may be asked to deliver presentations on aspects and extensions of the work. Chapter 1 is devoted to the relationship between random walks (on graphs) and electrical networks. This leads to the Thomson and Rayleigh principles, and thence to the proof of P´olya’s theorem. In Chapter 2, we describe Wilson’s algorithm for constructing a uniform spanning tree (UST), and we discuss boundary conditions and weak limits for UST on a lattice. This chapter includes a brief introduction to Schramm–L¨owner evolutions (SLE). Percolation theory appears first in Chapter 3, together with a short introduction to self-avoiding walks. Correlation inequalities and other general techniques are described in Chapter 4. A special feature of this part of the book is a fairly full c G. R. Grimmett 6 February 2009

viii

Preface

treatment of influence and sharp-threshold theorems. We return to percolation in Chapter 5, where the basic theory is summarised. There is a full account of Smirnov’s proof of Cardy’s formula. This is followed in Chapter 6 by a study of the contact model on lattices and trees. Chapter 7 begins with a proof of the equivalence of Gibbs states and Markov fields, and continues with an introduction to the Ising and Potts models. Chapter 8 is an account of the random-cluster model. The quantum Ising model features in the next chapter, particularly through its relationship to a continuum randomcluster model, and the consequent analysis using stochastic geometry. Interacting particle systems form the basis of Chapter 10. This is a large field in its own right, and little is done here beyond introductions to the contact, voter, and exclusion models. The chromatic number of a random graph features in Chapter 11 as an application of Hoeffding’s inequality for the tail of a martingale. The final Chapter 12 contains that most notorious open problem in stochastic geometry, the Lorentz model (or Ehrenfest wind–tree model) on the square lattice. These notes are based in part on courses given by the author within Part 3 of the Mathematical Tripos at Cambridge University over several years, and have been prepared in this form for the 2008 PIMS–UBC Summer School in Probability, and for the programme on Statistical Mechanics at the Institut Henri Poincar´e, Paris, during the last quarter of 2008. They have been written in part during a visit to the Mathematics Department at UCLA, to which the author expresses his gratitude for the warm welcome received there, and in part during programmes at the Isaac Newton Institute and the Institut Henri Poincar´e–Centre Emile Borel. The author thanks four artists for permission to include their work: Tom Kennedy (Fig. 2.1), Oded Schramm (Figs 2.2–2.4), Rapha¨el Cerf (Fig. 5.3), and Julien Dub´edat (Fig. 5.19). The section on influence has benefited from a conversation with Rob van den Berg. Stanislav Smirnov and Wendelin Werner have consented to the inclusion of some of their neat arguments, hitherto unpublished. Several readers have proposed suggestions and corrections. Thank you, everyone!

G. R. G. Cambridge January 2009

c G. R. Grimmett 6 February 2009

Contents

1

Random Walks on Graphs 1.1 1.2 1.3 1.4 1.5 1.6

2

3

4

1

Random walks and reversible Markov chains . . . . . . . . . . . . . . . . . . 1 Electrical networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Flows and energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Recurrence and resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 P´olya theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Uniform Spanning Tree

18

Percolation and Self-Avoiding Walk

33

Correlation and Concentration

44

2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wilson algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weak limits on lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniform forest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schramm–L¨owner evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-avoiding walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupled percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oriented percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Holley inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FKG inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BK inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hoeffding inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence for product measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proofs of influence theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Russo formula, and sharp thresholds . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

c G. R. Grimmett 6 February 2009

18 20 24 26 27 31

33 37 39 39 42

44 47 48 50 51 56 68 70

x

5

Contents

Further Percolation 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

6

7

8

9

Subcritical phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supercritical phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniqueness of the infinite cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open paths in annuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The critical probability in two dimensions . . . . . . . . . . . . . . . . . . . . Cardy formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The critical probability via the sharp-threshold theorem . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 78 83 86 89 92 99 108 112

Contact Model

114

6.1 6.2 6.3 6.4 6.5 6.6 6.7

114 115 117 119 121 124 127

Stochastic epidemics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupling and duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Invariant measures and percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . The critical value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The contact model on a tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Space–time percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gibbs States

128

7.1 7.2 7.3 7.4

128 130 133 135

Dependency graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Markov fields and Gibbs states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ising and Potts models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Random-Cluster Model

136

8.1 8.2 8.3 8.4 8.5 8.6 8.7

136 139 139 144 146 151 154

The random-cluster and Ising/Potts models. . . . . . . . . . . . . . . . . . . . Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infinite-volume limits and phase transition . . . . . . . . . . . . . . . . . . . . Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random even graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum Ising Model

157

9.1 9.2 9.3 9.4 9.5 9.6

157 158 160 165 166 169

The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuum random-cluster model . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Ising via random-cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . Long-range order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entanglement in one dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Interacting Particle Systems c G. R. Grimmett 6 February 2009

170

Contents

10.1 10.2 10.3 10.4 10.5

Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contact model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Voter model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exclusion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 Random Graphs 11.1 11.2 11.3 11.4

Erd˝os–R´enyi graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Giant component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Independence and colouring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 Lorentz Gas 12.1 12.2 12.3 12.4

Lorentz model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The square Lorentz gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

170 172 173 176 179

181 181 182 187 192

193 193 194 196 198

References

199

Index

212

c G. R. Grimmett 6 February 2009

1 Random Walks on Graphs

The theory of electrical networks is a fundamental tool for studying the recurrence of reversible Markov chains. The Kirchhoff laws and Thomson principle permit a neat proof of Po´ lya’s theorem for random walk on a ddimensional grid.

1.1 Random walks and reversible Markov chains Let G = (V, E) be a finite or countably infinite graph, which we assume for simplicity to have neither loops nor multiple edges. If G is infinite, we shall usually assume in addition that every vertex-degree is finite. A particle moves around the vertex-set V . Having arrived at the vertex Sn at time n, its next position Sn+1 is chosen uniformly at random from the set of neighbours of Sn . The trajectory of the particle is called a simple random walk (SRW) on G. Two of the basic questions concerning simple random walk are: 1. under what conditions is the walk recurrent, in that it returns (almost surely) to its starting point? 2. how does the distance between Sn and S0 behave as n → ∞? The above SRW is symmetric in that the jumps are chosen uniformly from the set of available neighbours. In a more general process, we take a function w : E → (0, ∞), and we jump along the edge e with probability proportional to we . Any reversible Markov chain on the set V gives rise to such a walk as follows. Let Z = (Z n : n ≥ 0) be a Markov chain on V with transition matrix P, and assume that Z is reversible with respect to some positive function π : V → (0, ∞), which is to say that (1.1)

πu pu,v = πv pv,u ,

u, v ∈ V.

With each distinct pair u, v ∈ V , we associate the weight (1.2)

c G. R. Grimmett 6 February 2009

wu,v = πu pu,v ,

2

[1.1]

Random Walks on Graphs

noting by (1.1) that wu,v = wv,u . Then (1.3)

pu,v =

where Wu =

wu,v , Wu X

u, v ∈ V,

wu,v ,

v∈V

u ∈ V.

That is, given that Z n = u, the chain jumps to a new vertex v with probability proportional to wu,v . This may be set in the context of a random walk on the graph with the vertex-set V , and with edge-set containing all e = hu, vi such that pu,v > 0. With the edge e we associate the weight we = wu,v . In this chapter, we develop the relationship between random walks on G and electrical networks on G. There are some excellent accounts of this area already, and the reader is referred to the books of Doyle and Snell [66] and Lyons and Peres [157], amongst others. The connection between these two topics is made via the so-called ‘harmonic functions’ of the random walk. (1.4) Definition. Let U ⊆ V , and let Z be a Markov chain on V with transition matrix P, that is reversible with respect to the positive function π . The function f : V → R is harmonic (for Z ) on U if f (u) =

X

v∈V

pu,v f (v),

u ∈ U,

or equivalently, if f (u) = E( f (Z 1 ) | Z 0 = u) for u ∈ U .

From the pair (P, π ), one can construct the graph G as above, and the weight function w as in (1.2). We refer to the pair (G, w) as the weighted graph associated with (P, π ). We shall speak of f as being harmonic (for (G, w)) on U if it is harmonic for Z on U . The hitting probabilities are the basic examples of harmonic functions for the chain Z . Let W ⊆ V and s ∈ / W . For u ∈ U = V \ W , let g(u) be the probability that the chain, started from u, hits s before W , that is, g(u) = Pu (Z n = s for some n < TW ), where

TW = inf{n ≥ 0 : Z n ∈ W }

is the first passage time to W , and Pu (·) = P(· | Z 0 = u). (1.5) Theorem. The function g is harmonic on U \ {s}. Evidently, g(s) = 1, and g(v) = 0 for v ∈ W . We speak of these values of g as being the ‘boundary conditions’ of the harmonic function g. c G. R. Grimmett 6 February 2009

[1.2]

Electrical networks

3

Proof. This is an elementary exercise using the Markov property. For u ∈ / W ∪{s}, g(u) = =

X

v∈V

X

pu,v Pu (Z n = s for some n < TW | Z 1 = v) pu,v g(v),

v∈V

as required.



1.2 Electrical networks Throughout this section, G = (V, E) is a finite graph with neither loops nor multiple edges, and w : E → (0, ∞) is a weight function on the edges. We shall assume further that G is connected. We may build an electrical network with diagram G, in which the edge e has conductance we (or, equivalently, resistance 1/we ). Let s, t ∈ V be distinct vertices termed the source and sink, and suppose we connect a battery across the pair s, t. It is a physical observation that electrons flow along the wires in the network. The flow is described by the so-called Kirchhoff laws, as follows. To each edge e = hu, vi, there are associated (directed) quantities φ u,v and i u,v , called the potential difference from u to v, and the current from u to v, respectively. These are antisymmetric, φu,v = −φv,u ,

i u,v = −i v,u .

(1.6) Kirchhoff’s potential law. The cumulative potential difference around any cycle x 1 , x 2 , . . . , x n , x n+1 = x 1 of G is zero, that is, n X

(1.7)

j =1

φx j ,x j +1 = 0.

(1.8) Kirchhoff’s current law. The total current flowing out of any vertex u ∈ V other than the source and sink is zero, that is, (1.9)

X

v∈V

i u,v = 0,

u 6= s, t.

The relationship between resistance/conductance, potential difference, and current is given by Ohm’s law. c G. R. Grimmett 6 February 2009

4

[1.2]

Random Walks on Graphs

(1.10) Ohm’s law. For any edge e = hu, vi, i u,v = we φu,v . Kirchhoff’s potential law is equivalent to the statement that there exists a function φ : V → R, called a potential function, such that φu,v = φ(v) − φ(u),

hu, vi ∈ E.

Since φ is determined up to an additive constant, we are free to pick the potential of any single vertex. Note the convention that current flows uphill: i u,v has the same sign as φu,v = φ(v) − φ(u). (1.11) Theorem. A potential function is harmonic on the set of vertices other than the source and sink. Proof. Let U = V \ {s, t}. By Kirchhoff’s current law and Ohm’s law, X

v∈V

wu,v [φ(v) − φ(u)] = 0,

u ∈ U,

which is to say that X wu,v φ(v), Wu

φ(u) =

v∈V

where Wu = That is, φ is harmonic on U .

X

u ∈ U,

wu,v .

v∈V



We can use Ohm’s law to express the potential differences in terms of the currents, and thus the two Kirchhoff laws may be viewed as concerning the currents only. Relation (1.7) becomes (1.12)

n X i x j ,x j +1 j =1

whx j ,x j +1 i

= 0,

valid for any cycle x 1 , x 2 , . . . , x n , x n+1 = x 1 . With (1.7) written thus, each law is linear in the currents, and the superposition principle follows. (1.13) Theorem. Superposition principle. If i 1 and i 2 are solutions of the two Kirchhoff laws, then so is the sum i 1 + i 2 . Next we introduce the concept of a ‘flow’ on the graph.

c G. R. Grimmett 6 February 2009

[1.2]

Electrical networks

5

(1.14) Definition. Let s, t ∈ V , s 6= t. An s/t-flow j is a vector j = ( ju,v : u, v ∈ V, u 6= v), such that (i) ju,v = − jv,u , (ii) ju,v = 0 whenever u  v, P (iii) for any u 6= s, t, we have that v∈V ju,v = 0.

The vertices s amd t are called the ‘source’ and ‘sink’ of an s/t-flow, and we usually abbreviate ‘s/t-flow’ to ‘flow’. For any flow j , we write X Ju = ju,v , u ∈ U, v∈V

noting by (iii) above that Ju = 0 for u 6= s, t. Thus, X X X ( ju,v + jv,u ) = 0. Js + J t = ju = ju,v = 12 u∈V

u,v∈V

u,v∈V

Therefore, Js = −Jt , and we call |Js | the size of the flow j , denoted | j |. If |Js | = 1, we call j a unit flow. We shall normally take Js > 0, in which case s is the source, and t the sink. (1.15) Theorem. Let i 1 and i 2 be two solutions of the Kirchhoff laws with equal size. Then i 1 = i 2 .

Proof. By the superposition principle, j = i 1 −i 2 satisfies the two Kirchhoff laws. Furthermore, under the flow j , no current enters or leaves the system. Therefore, Jv = 0 for all v ∈ V . Suppose ju 1 ,u 2 > 0 for some edge hu 1 , u 2 i. By the Kirchhoff current law, there exists u 3 such that ju 2 ,u 3 > 0. By iteration, there exists a cycle u 1 , u 2 , . . . , u n , u n+1 = u 1 such that ju j ,u j +1 > 0 for j = 1, 2, . . . , n. By Ohm’s law, the corresponding potential function satisfies φ(u 1 ) < φ(u 2 ) < · · · < φ(u n+1 ) = φ(u 1 ), a contradiction. Therefore, ju,v = 0 for all u, v.



For a given size of input current, there can be no more than one solution to the two Kirchhoff laws, but is there a solution at all? The answer can be expressed in terms of counts of spanning trees. Consider first the special case when w e = 1 for all e ∈ E. Let N be the number of spanning trees of G. For any edge ha, bi, let 5(s, a, b, t) be the property of spanning trees that: the unique s/t path in the tree passes along the edge ha, bi in the direction from a to b. Let N (s, a, b, t) be the set of spanning trees of G with the property 5(s, a, b, t), and N (s, a, b, t) = |N (s, a, b, t)|. (1.16) Theorem. The function (1.17)

i a,b =

 1 N (s, a, b, t) − N (s, b, a, t) , N

c G. R. Grimmett 6 February 2009

ha, bi ∈ E,

6

[1.2]

Random Walks on Graphs

defines a flow of size 1 that satisfies the Kirchhoff laws. Let T be a spanning tree of G chosen uniformly at random from the set T of all such spanning trees. By the theorem and the previous discussion, the unique solution to the Kirchhoff laws with size 1 is given by

i a,b = P T has 5(s, a, b, t) − P T has 5(s, b, a, t) .

We shall return to uniform spanning trees in Chapter 2. We prove Theorem 1.16 next. Exactly the same proof is valid in the case of general conductances we . In that case, we define the weight of a spanning tree T as Y w(T ) = we , e∈T

and we set (1.18)

N∗ =

X

N ∗ (s, a, b, t) =

w(T ),

T ∈T

X

w(T ).

T with 5(s,a,b,t)

The conclusion of Theorem 1.16 holds in this setting with i a,b =

 1  ∗ ∗ N (s, a, b, t) − N (s, b, a, t) , N∗

ha, bi ∈ E,

Proof of Theorem 1.16. We first check the Kirchhoff current law. In every spanning tree T , there exists a unique vertex b such that the s/t path of T contains the edge hs, bi, and the path traverses this edge from s to b. Therefore, X

b∈V

N (s, s, b, t) = N ,

By (1.17), P

X

b∈V

N (s, b, s, t) = 0 for b ∈ V.

i s,b = 1,

and, by a similar argument, b∈V i b,t = 1. Let T be a spanning tree of G. The contribution towards the quantity i a,b , made by T , depends on the s/t path π of T , and equals N −1 (1.19)

−N

−1

0

if π passes along ha, bi from a to b, if π passes along ha, bi from b to a,

if π does not contain the edge ha, bi.

P Let v ∈ V , v 6= s, t, and write Iv = w∈V i v,w . If v ∈ π , the contribution of T towards i v is N −1 − N −1 = 0, since π arrives at v along some edge of the c G. R. Grimmett 6 February 2009

[1.3]

Flows and energy

7

form ha, vi, and departs v along some edge of the form hv, bi. If v ∈ / π , then T contributes 0 to Iv . Summing over T , we obtain that Iv = 0 for all v 6= s, t, as required. We next check the Kirchhoff potential law. Let x 1 , x 2 , . . . , x n , x n+1 = x 1 be a cycle C of G. We shall show that n X

(1.20)

j =1

i x j ,x j +1 = 0,

and this will confirm (1.12), on recalling that we = 1 for all e ∈ E. It is more convenient in this context to work with ‘bushes’ than spanning trees. A bush is defined to be a forest on V containing exactly two trees, one denoted Ts and containing s, and the other denoted Tt and containing t. We write (Ts , Tt ) for this bush. Let e = ha, bi, and let B(s, a, b, t) be the set of bushes with a ∈ Ts and b ∈ Tt . The sets B(s, a, b, t) and N (s, a, b, t) are in one–one correspondence, since the addition of e to a B ∈ B(s, a, b, t) creates a unique member T = T (B) of N (s, a, b, t), and vice versa. By (1.19) and the above, a bush B = (Ts , Tt ) makes a contribution to i a,b of: N −1 −N −1

0

if B ∈ B(s, a, b, t),

if B ∈ B(s, b, a, t),

otherwise.

Therefore, B makes a contribution towards the sum in (1.20) that is equal to N −1 (F+ − F− ), where F+ (respectively, F− ) is the number of pairs x j , x j +1 of C, 1 ≤ j ≤ n, with x j ∈ Ts , x j +1 ∈ Tt (respectively, x j +1 ∈ Ts , x j ∈ Tt ). Since C is a cycle, F+ = F− , whence each bush contributes 0 to the sum, and (1.20) is proved. 

1.3 Flows and energy Let G = (V, E) be a connected graph as before, and let s, t ∈ V be distinct vertices. Let j be an s/t-flow. With we the conductance of the edge e, the (dissipated) energy of j is defined to be E( j ) =

X

e=hu,vi∈E

2 /we = ju,v

1 2

X

u,v∈V u∼v

The following piece of linear algebra will be useful. c G. R. Grimmett 6 February 2009

2 /whu,vi ju,v

8

[1.3]

Random Walks on Graphs

e

u

v

w

e

f

f Figure 1.1. Two edges e and f in parallel and in series.

(1.21) Proposition. Let ψ : V → R, and let j be an s/t-flow. Then X [ψ(v) − ψ(u)] ju,v . [ψ(t) − ψ(s)]Js = 21 u,v∈V

Proof. By the properties of a flow, X X X [ψ(v) − ψ(u)] ju,v = ψ(v)(−Jv ) − ψ(u)Ju u,v∈V

v∈V

u∈V

= −2[ψ(s)Js + ψ(t)Jt ] = 2[ψ(t) − ψ(s)]Js ,

as required.



Let φ and i satisfy the Kirchhoff laws. We apply Proposition 1.21 with ψ = φ and j = i to find by Ohm’s law that (1.22)

E(i ) = [φ(t) − φ(s)]i s .

That is, the energy of the true current-flow i between s to t equals the energy dissipated in a single hs, ti edge carrying the same potential difference and total current. The conductance Weff of such an edge would satisfy Ohm’s law, that is, (1.23)

i s = Weff [φ(t) − φ(s)],

and we define the effective conductance Weff by this equation. The effective resistance is (1.24)

Reff =

1 , Weff

which, by (1.22)–(1.23), equals E(i )/i s2 . We state this as a lemma. (1.25) Lemma. The effective resistance Reff of the network between vertices s and t equals the dissipated energy when a unit flow passes from s to t. It is useful to be able to do calculations. Electrical engineers have devised a variety of formulaic methods for calculating the effective resistance of a network, of which the simplest are the series and parallel laws, illustrated in Figure 1.1. (1.26) Series law. Two resistors of size r 1 and r2 in series may be replaced by a single resistor of size r1 + r2 . c G. R. Grimmett 6 February 2009

[1.3]

Flows and energy

9

(1.27) Parallel law. Two resistors of size r 1 and r2 in parallel may be replaced by a single resistor of size R where R −1 = r1−1 + r2−1 .

A third such rule, the so-called ‘star–triangle transformation’, may be found in Exercise 1.5.

(1.28) Theorem. Thomson principle. Let G = (V, E) be a connected graph, and we , e ∈ E, (strictly positive) conductances. Let s, t ∈ V , s 6= t. Amongst all unit flows through G from s to t, the flow that satisfies the Kirchhoff laws is the unique s/t-flow i that minimizes the dissipated energy. That is,  E(i ) = inf E( j ) : j a unit flow from s to t . Proof. Let j be a unit flow from source s to sink t, and set k = j − i where i is the (unique) unit-flow solution to the Kirchhoff laws. Thus, k is a flow with zero size. Now, with e = hu, vi and r e = 1/we , 2E( j ) = =

X

u,v∈V

X

2 re = ju,v 2 re ku,v

u,v∈V

+

X

u,v∈V

(ku,v + i u,v )2 re

X

u,v∈V

2 re + 2 i u,v

X

i u,v ku,v re .

u,v∈V

Let φ be the potential function corresponding to i . By Ohm’s law and Proposition 1.21, X

u,v∈V

i u,v ku,v re =

X

u,v∈V

[φ(v) − φ(u)]ku,v

= 2[φ(t) − φ(s)]K s ,

which equals zero. Therefore E( j ) ≥ E(i ), with equality if and only if j = i .  The Thomson ‘variational principle’ leads to a proof of the ‘obvious’ fact that the effective resistance of a network is a non-decreasing function of the resistances of individual edges. (1.29) Theorem. Rayleigh principle. The effective resistance Reff of the network is a non-decreasing function of the edge-resistances (r e : e ∈ E). It is left as an exercise to show that Reff is a concave function of the (r e ). See Exercise 1.6. Proof. Consider two sets (r e : e ∈ E) and (re0 : e ∈ E) of edge-resistances such that re ≤ re0 for all e. Let i and i 0 denote the corresponding unit flows satisfying c G. R. Grimmett 6 February 2009

10

[1.4]

Random Walks on Graphs

the Kirchhoff laws. Then, with r e = rhu,vi , Reff =

1 2

2 i u,v re

u,v∈V u∼v

≤

1 2

≤

1 2

=

X

X

0 )2 r e (i u,v

by the Thomson principle

0 )2 re0 (i u,v

since re ≤ re0

u,v∈V u∼v

X

u,v∈V u∼v 0 Reff ,

as required.



1.4 Recurrence and resistance Let G = (V, E) be an infinite connected graph with finite vertex degrees, and let (we : e ∈ E) be (strictly positive) conductances. We shall consider a reversible Markov chain Z = (Z n : n ≥ 0) on V with transition probabilities given by (1.3). Our purpose is to establish a condition on the pair (G, w) that is equivalent to the recurrence of Z . Let 0 be a distinguished vertex of G, called the ‘origin’, and suppose Z 0 = 0. The graph-theoretic distance between two vertices u, v is the number of edges in the shortest path between u and v, denoted d(u, v). Let 3n = {u ∈ V : d(0, v) ≤ n},

∂3n = 3n \ 3n−1 = {u ∈ V : d(0, v) = n}.

For n ≥ 1, we let G n be the graph obtained from G by identifying all vertices in V \ 3n−1 , and we denote the identified vertex as In . The resulting finite graph G n may be considered as an electrical network with source 0 and sink I n . Let Reff (n) be the effective resistance of this network. The graph G n may be obtained from G n+1 by identifying all vertices lying in ∂3n ∪ ∂3n+1 , and thus, by the Rayleigh principle, Reff (n) is non-decreasing in n. Therefore the limit Reff = lim Reff (n) n→∞

exists. (1.30) Theorem. The probability of return by Z to 0 is given by P0 (Z n = 0 for some n ≥ 1) = 1 − c G. R. Grimmett 6 February 2009

1 , W0 Reff

[1.4]

Recurrence and resistance

11

e 0

Figure 1.2. This is an infinite binary tree with two parallel edges joining the origin to the root. When each edge has unit resistance, it is an easy calculation that R eff = 23 , so the probability of return to 0 is 32 . If the edge e is removed, this probability becomes 21 .

where W0 =

P

v: v∼0 wh0,vi .

The return probability is non-decreasing if W0 Reff is increased. By the Rayleigh principle, this can be achieved, for example, by removing an edge of E that is not incident to 0. The removal of an edge incident to 0 can have the opposite effect, since W0 decreases while Reff increases. See Figure 1.2. (1.31) Corollary. (a) The chain Z is recurrent if and only if Reff = ∞.

(b) The chain Z is transient if and only if there exists a non-zero flow G P j on 2 /w from 0 to ∞ (that is, there is no sink) whose energy E( j ) = j e e e satisfies E( j ) < ∞. It is left as an exercise to extend this to countable graphs G with unbounded degrees and satisfying Wu < ∞ for every vertex u. Proof of Theorem 1.30. Let gn (v) = Pv (Z hits ∂3n before 0),

v ∈ 3n .

By Theorem 1.5, gn is the unique harmonic function on G n with boundary conditions gn (0) = 0,

gn (v) = 1 for v ∈ ∂3n .

Therefore, gn is a potential function on G n viewed as an electrical network with source 0 and sink In . c G. R. Grimmett 6 February 2009

12

[1.4]

Random Walks on Graphs

By conditioning on the first step of the walk, and using Ohm’s law, X P0 (Z returns to 0 before reaching ∂3n ) = 1 − p0,v gn (v) v: v∼0

=1−

=1−

X w0,v [gn (v) − gn (0)] W0

v: v∼0

|i (n)| , W0

where i (n) is the flow of currents in G n , and |i (n)| is its size. By (1.23)–(1.24), |i (n)| = 1/Reff (n). The theorem is proved on noting that P0 (Z returns to 0 before reaching ∂3n ) → P0 (Z n = 0 for some n ≥ 1)

as n → ∞, by the continuity of probability measures.



Proof of Corollary 1.31. Part (a) is an immediate consequence of Theorem 1.30, and we turn to part (b). By Lemma 1.25, there exists a unit flow i (n) in G n , with source 0 and sink ∂3n , and with energy E(i (n)) = Reff (n). Let i be a non-zero 0/∞-flow; by normalizing by its size, we may take i to be a unit flow. When restricted to the edge-set E n of 3n , i forms a unit flow from 0 to ∂3n . By the Thomson principle, Theorem 1.28, X E(i n ) ≤ i e2 /we ≤ E(i ), e∈E n

whence,

E(i ) ≥ lim E(i n ) = Reff . n→∞

Therefore, by (a), E(i ) = ∞ if the chain is transient. Suppose, conversely, that the chain is recurrent. By diagonal selection, there exists a subsequence (n k ) along which i (n k ) converges to some limit i . Since each i (n k ) is a unit flow, i is a unit 0/∞-flow. Now, X i (n k )2e /we E(i (n k )) = e∈E

≥

X

e∈E m

→

Therefore,

i (n k )2e /we

X

e∈E m

→ E(i )

i (e)2 /we

as k → ∞ as m → ∞.

E(i ) ≤ lim Reff (n k ) = Reff < ∞, k→∞

and i is the required flow. c G. R. Grimmett 6 February 2009



[1.5]

P´olya theorem

0

1

2

13

3

Figure 1.3. The vertex labelled i is a composite vertex obtained by identifying all vertices with distance i from 0. There are 8i − 4 edges joining vertices i − 1 and i .

1.5 P´olya theorem The following celebrated theorem 1 can be be proved by estimating effective resistances. (1.32) Theorem [175]. Symmetric random walk on Zd is recurrent if d = 1, 2 and transient if d ≥ 3.

The advantage of the following proof of P´olya’s theorem over more standard arguments is its robustness with respect to the underlying graph. Similar arguments are valid for graphs that are, in broad terms, comparable to Zd when viewed as electrical networks. Proof. For simplicity, and with only little loss of generality, we shall concentrate on the cases d = 2, 3. Let d = 2, for which case we aim to show that R eff = ∞. This is achieved by finding an infinite lower bound for Reff , and lower bounds can be obtained by decreasing individual edge-resistances. The identification of two vertices of a network amounts to the addition of a resistor with 0 resistance, and, by the Rayleigh principle, the effective resistance of the network can only decrease. From Z2 , we construct a new graph in which, for each k = 1, 2, . . . , the set ∂3k = {v ∈ Z2 : d(0, v) = k} is identified as a singleton. This transforms Z2 into the graph shown in Figure 1.3. By the series/parallel laws and the Rayleigh principle, n−1 X 1 Reff (n) ≥ , 8i − 4 i =1

whence Reff (n) ≥ c log n → ∞ as n → ∞. Suppose now that d = 3. There are at least two ways of proceeding. We shall present one such route from [158], and we shall then sketch the second inspired by [66]. By the remark after Theorem 1.31, it suffices to construct a non-zero flow from 0 with finite energy, and we proceed to do this. Let S be the surface of the unit sphere of R3 with centre at the origin 0. Take u ∈ Z3 , u 6= 0, and position a unit cube C u in R3 with centre at u; see Figure 1.4. For each neighbour v of u, the directed edge [u, vi intersects a unique face, denoted Fu,v , of C u . For x ∈ R3 , x 6= 0, let 5(x) be the point of intersection with S of the straight line segment from 0 to x. Let ju,v be equal in absolute value to the surface measure 1 An

amusing story is told in [176] about Po´ lya’s inspiration for this theorem.

c G. R. Grimmett 6 February 2009

14

[1.5]

Random Walks on Graphs

Cu

Fu,v

S

5(Fu,v ) 0

Figure 1.4. The flow along the edge hu, vi is equal to the area of the projection 5(Fu,v ) on the unit sphere centred at the origin.

of 5(Fu,v ). The sign of ju,v is taken to be positive if and only if the dot product of 21 (u + v) and v − u, viewed as vectors in R3 , is positive. Let jv,u = − ju,v . We claim that j is a flow on Z3 . Parts (i) and (ii) of Definition 1.14 follow by construction, and it remains to check (iii). P The surface of C u has a projection 5(C u ) on S. The sum Ju = v∼u ju,v is the integral over x ∈ 5(C u ), with respect to surface measure, of the number of neighbours v of u (counted with sign) for which x ∈ 5(Fu,v ). Almost every x ∈ 5(C u ) is counted twice, with signs + and −. Thus the integral equals 0, whence Ju = 0 for all u 6= 0. It is easily seen that j0 6= 0, so j is a non-zero flow. Next we estimate its energy. By an elementary geometric consideration, there exist ci < ∞ such that: (a) | ju,v | ≤ c1 /|u|2 for u 6= 0, where |u| = d(0, u) is the length of the shortest path from 0 to u, (b) the number of u ∈ Z3 with |u| = n is smaller than c2 n 2 . It follows that E( j ) ≤

XX u6=0 v∼u

as required.

2 ≤ ju,v

∞ X n=1

6c2 n 2

 c 2 1

n2

< ∞, 

Another way of showing Reff < ∞ is to find a finite upper bound for Reff . Upper bounds can be obtained by increasing individual edge-resistances, or by removing edges. The idea is to embed a tree with finite resistance in Z 3 . Consider a binary tree Tρ in which the connections between generation n − 1 and generation n have resistance ρ n , where ρ > 0. It is an easy exercise using the series/parallel c G. R. Grimmett 6 February 2009

[1.6]

Exercises

15

laws that the effective resistance between the root and infinity is Reff (Tρ ) =

∞ X

(ρ/2)n ,

n=1

which we make finite by choosing ρ < 2. We proceed to embed Tρ in Z3 in such a way that a connection between generation n − 1 and generation n is a lattice-path of length order ρ n . There are 2n vertices of Tρ in generation n, and their latticeP distance from 0 has order nk=1 ρ k , that is, order ρ n . The surface of the k-ball in R3 has order k 2 , and thus it is necessary that c(ρ n )2 ≥ 2n ,

√ which is to say that ρ > 2. √ Let 2 < ρ < 2. It is now fairly simple to check that Reff < c0 Reff (Tρ ). This method has been used in [102] to prove the transience of the infinite open cluster of percolation on Z3 . It is related to, but different from, the tree embeddings of [66].

1.6 Exercises 1.1. Let G = (V, E) be a finite connected graph with unit edge-weights. Show that the effective resistance between two nodes s, t of the associated electrical network may be expressed as B/N , where B is the number of bushes of G, and N is the number of its spanning trees. (See the proof of Theorem 1.16.) Extend this result to general positive edge-weights we . 1.2. Let G = (V, E) be a finite connected graph with positive edge-weights (we : e ∈ E), and let N ∗ be given by (1.18). Show that i a,b =

 1  ∗ N (s, a, b, t) − N ∗ (s, b, a, t) ∗ N

constitutes a unit flow through G from s to t satisfying Kirchhoff’s laws. 1.3. (continuation) Let G = (V, E) be finite and connected with P given conductances (we : e ∈ E), and let (x v : v ∈ V ) be reals satisfying v x v = 0. To G we append a notional vertex labelled ∞, and we join ∞ to each v ∈ V . Show that there exists a solution i to Kirchhoff’s laws on the expanded graph, viewed as two laws concerning current flow, such that the current along the edge hv, ∞i is x v . 1.4. Prove the series and parallel laws for electrical networks. 1.5. Star–triangle transformation. The triangle T is replaced by the star S in an electrical network, as illustrated in Figure 1.5. Explain the sense in which the c G. R. Grimmett 6 February 2009

16

[1.6]

Random Walks on Graphs

C

C r10

r3 r2 r20 A

B

r1

A

r30 B

Figure 1.5. Edge-resistances in the star–triangle transformation. The triangle T on the left is replaced by the star S on the right, and the corresponding resistances are as marked.

two networks are the same, when the resistances are chosen such that r j r j0 = c for j = 1, 2, 3 and some constant c to be determined. 1.6. Let R(r ) be the effective resistance between two given vertices of a finite network with edge-resistances r = (r (e) : e ∈ E). Show that R is concave in that 1 2



 R(r1 ) + R(r2 ) ≤ R

1 2 (r1


+ r2 ) .

1.7. Maximum principle. Let G = (V, E) be a finite or infinite network with associated conductances (we : e ∈ E), and let H = (W, F) be a connected subgraph of G. Let φ : V → [0, ∞) be harmonic on W , and suppose the supremum of φ on W is achieved and satisfies sup φ(w) = kφk∞ := sup φ(v).

w∈W

v∈V

Show that φ is constant on W ∪ ∂ W , and equals k f k∞ . 1.8. Let G be an infinite connected graph, and let ∂3n be the set of vertices distance n from the vertex labelled 0. With E n the number P of edges joining ∂3n to ∂3n+1 , show that random walk on G is recurrent if n E n−1 = ∞. 1.9. (continuation) Assume that G is ‘spherically symmetric’ in that: for all n, for all x, y ∈ ∂3n , there exists a graph automorphism that fixes 0 and maps x P to y. Show that random walk on G is transient if n E n−1 < ∞. 1.10. Let G be a finite connected network with positive conductances (w e : e ∈ E), and let a, b be distinct vertices. Let i x y denote the current along an edge from x to y when a unit current flows from the source vertex a to the sink vertex b. Run the associated Markov chain, starting at a, until it reaches b for the first time, and let u x,y be the mean of the total number of transitions of the chain between x and y. Transitions from x to y count positive, and from y to x negative, so that u x,y is the mean number of transitions from x to y, minus the mean number from y to x. Show that i x,y = u x,y . 1.11. Consider Z2 as an electrical network with unit resistances, and suppose we identify all vertices that are distance n or more from the origin. Show that the resistance between the origin and the composite vertex is at most C log n for some C < ∞. c G. R. Grimmett 6 February 2009

2 Uniform Spanning Tree

The Uniform Spanning Tree (UST) measure has a property of negative correlation. A similar property is conjectured for Uniform Forest and Uniform Connected Subgraph. Wilson’s algorithm is an efficient way to construct a UST. The UST on the infinite square grid may be defined as the weak limit of the finite-volume measures, and it converges in a certain manner to SLE 8 as the grid size approaches zero.

2.1 Definition Let G = (V, E) be a finite connected graph, and write T for the set of all spanning trees of G. Let T be picked uniformly at random from T . We call T a uniform spanning tree, abbreviated to UST. It is governed by the uniform measure: P(T = t) =

1 , |T |

t ∈T.

We may think of T either as a random graph, or as a random subset of E. In the latter case, T may be thought of as a random element of the set  = {0, 1} E of 0/1 vectors indexed by E. It is fundamental that UST has a property of negative correlation. In it simplest form, this may be expressed as follows. (2.1) Theorem. For f, g ∈ E, f 6= g, (2.2)

P( f ∈ T | g ∈ T ) ≤ P( f ∈ T ).

The proof makes striking use of the Thomson Principle via the monotonicity of effective resistance. One obtains the following by a mild extension of the proof. For B ⊆ E and g ∈ E \ B, (2.3)

P(B ⊆ T | g ∈ T ) ≤ P(B ⊆ T ).

c G. R. Grimmett 6 February 2009

18

[2.1]

Uniform Spanning Tree

Proof. Consider G as an electrical network each of whose edges have resistance 1. Let e = hx, yi, and denote by i = (i v,w : v, w ∈ V ) the current flow in G when a unit current enters at x and leaves at y. By Theorem 1.16, i x,y =

N (x, x, y, y) N

where N (x, x, y, y) is the number of spanning trees of G whose unique x/y path passes along the edge e in the direction from x to y, and N = |T |. Therefore, i x,y = P(e ∈ T ). Since hx, yi has unit resistance, i x,y equals the potential difference φ(y) − φ(x). By (1.22), (2.4)

G P(e ∈ T ) = Reff (x, y),

the effective resistance of G between x and y. Let f , g be distinct edges, and write G.g for the graph obtained from G by contracting g to a single vertex. There is a one–one correspondence between spanning trees of G containing g, and spanning trees of G.g. Therefore, P( f ∈ T | g ∈ T ) is simply the proportion of spanning trees of G.g that contain f . By (2.4), G.g P( f ∈ T | g ∈ T ) = Reff (x, y).

By the Rayleigh principle, Theorem 1.29, G.g

G Reff (x, y) ≤ Reff (x, y),

and the theorem is proved.



Theorem 2.1 has been extended by Feder and Mihail [78] to more general ‘increasing’ events. Let  = {0, 1} E , the set of 0/1 vectors indexed by E, and denote by ω = (ω(e) : e ∈ E) a typical member of . The partial order ≤ on  is the usual pointwise ordering: ω ≤ ω 0 if ω(e) ≤ ω0 (e) for all e ∈ E. A subset A ⊆  is called increasing if: for all ω, ω 0 ∈  satisfying ω ≤ ω0 , we have that ω0 ∈ A whenever ω ∈ A. For A ⊆  and F ⊆ E, we say that A is defined on F if A = C × {0, 1} E \F for some C ⊆ {0, 1} F . We refer to F as the ‘base’ of the event A. (2.5) Theorem [78]. Let F ⊆ E, and let A and B be increasing subsets of  such that: A is defined on F, and B is defined on E \ F. Then P(T ∈ A | T ∈ B) ≤ P(T ∈ A). Theorem 2.1 is retrieved by setting A = {ω ∈  : ω( f ) = 1} and B = {ω ∈  : ω(g) = 1}. The original proof of Theorem 2.5 is set in the context of matroid theory, and a further proof may be found in [29]. Whereas ‘positive correlation’ is well developed and understood as a technique for studying interacting systems, ‘negative correlation’ possesses some inherent difficulties. See [173] for further discussion. c G. R. Grimmett 6 February 2009

[2.2]

Wilson algorithm

19

2.2 Wilson algorithm There are various ways to generate a uniform spanning tree (UST) of the graph G. The following method, called Wilson’s algorithm [213], highlights the close relationship between UST and random walk. Take G = (V, E) to be a finite connected graph. We shall perform random walks on G subject to a process of so-called loop-erasure that we describe next 1 . Let W = (w0 , w1 , . . . , wk ) be a walk on G, which is to say that wi ∼ wi +1 for 0 ≤ i < k (note that the walk may have self-intersections). From W we construct a non-self-intersecting sub-walk, denoted LE(W ), by the removal of loops as they occur. More precisely, let J = min{ j ≥ 1 : w j = wi for some i < j }, and let I be the unique value of i satisfying I < J and w I = w J . Let W 0 = (w0 , w1 , . . . , w I , w J +1 , . . . , wk ) be the sub-walk of W obtained through the removal of the cycle (w I , w I +1 , . . . , w J ). This operation of single-loop-removal is iterated until no loops remain, and we denote by LE(W ) the surviving path from w0 to wk . Here is Wilson’s algorithm. First, we order the vertex-set V = (v1 , v2 , . . . , vn ) in an arbitrary manner. 1. Perform a random walk on G beginning at vi 1 with i 1 = 1, and stopped at the first time it visits vn . The outcome is a walk W1 = (u 1 = v1 , u 2 , . . . , u r = vn ). 2. From W1 we obtain the loop-erased path LE(W1 ), joining v1 to vn and containing no loops2 . Set T1 = LE(W1 ). 3. Find the earliest vertex, vi 2 say, of V not belonging to T1 , and perform a random walk beginning at vi 2 , and stopped at the first moment it hits some vertex of T1 . Call the resulting walk W2 , and loop-erase W2 to obtain some non-self-intersecting path LE(W2 ) from vi 2 to T1 . Set T2 = T1 ∪ LE(W2 ), the union of two edge-disjoint paths. 4. Iterate the above process, by running and loop-erasing a random walk from / Tj until it strikes the set Tj previously constructed. a new vertex vi j +1 ∈ 5. Stop when all vertices have been visited, and set T = T N , the final value of the Tj . Each stage of the above algorithm results in a sub-tree of G. The final such sub-tree T is spanning since, by assumption, it contains every vertex of V . (2.6) Theorem [213]. The graph T is a uniform spanning tree on G. Note that the initial ordering of V plays no role in the law of T . 1 Graph

theorists might prefer to call this cycle-erasure. and then erase its loops, the outcome is called loop-erased random walk, often abbreviated to LERW. 2 If we run a random walk

c G. R. Grimmett 6 February 2009

20

[2.2]

Uniform Spanning Tree

There are of course other ways of generating a UST on G, and we mention the well-known Aldous–Broder algorithm, [17, 48], that proceeds as follows. Choose a vertex r of G and perform a random walk on G, starting at r , until every vertex has been visited. For w ∈ V , w 6= r , let [v, wi be the directed edge that is traversed by the walk on its first visit to w. The edges thus obtained, when undirected, constitute a uniform spanning tree. The Aldous–Broder algorithm is closely related to the Wilson algorithm via a certain reversal of time, see [178]. We present the proof of Theorem 2.6 in a more general setting than UST. Heavy use will be made of [157] and the concept of ‘cycle popping’ introduced in the original paper [213] of David Wilson. Of considerable interest is an analysis of the run-time of Wilson’s algorithm, see [178]. Consider an irreducible Markov chain with transition matrix P on the finite state space S. With this chain we may associate a directed graph H = (S, F) much as in Section 1.1. This graph H has vertex-set S, and edge-set F = {[x, yi : p x,y > 0}. We refer to x (respectively, y) as the head (respectively, tail) of the (directed) edge e = [x, yi, written x = e− , y = e+ . Since the chain is irreducible, H is connected in the sense that, for all x, y ∈ S, there exists a directed path from x to y. Let r ∈ S be a distinguished vertex called the root. A spanning arborescence of H with root r is a subgraph A with the following properties: (a) each vertex of S apart from r is the head of a unique edge of A, (b) the root r is the head of no edge of A, (c) A possesses no (directed) cycles. S Let 6r be the set of all spanning arborescences with root r , and 6 = r ∈S 6r . It is easily seen that there exists a unique (directed) path in the spanning arborescence A joining any given vertex x to the root. To the spanning arborescence A we assign the weight (2.7)

α( A) =

Y

e∈A

pe− ,e+ ,

and we shall describe a randomized algorithm that selects a given spanning arborescence A with probability proportional to α( A). Since α( A) is independent of the diagonal elements pz,z of P, and each x (6= r ) is the head of a unique edge of A, we may assume that pz,z = 0 for all z ∈ S. Let r ∈ S. Wilson’s algorithm is easily adapted in order to sample from 6r . Let v1 , v2 , . . . , vn−1 be an ordering of S \ {r }. 1. Let σ0 = {r }. 2. Sample a Markov chain with transition matrix P beginning at vi 1 with i 1 = 1, and stopped at the first time it hits σ0 . The outcome is a (directed) walk W1 = (u 1 = v1 , u 2 , . . . , u k , r ). From W1 we obtain the loop-erased path σ1 = LE(W1 ), joining v1 to r and containing no loops. 3. Find the earliest vertex, vi 2 say, of S not belonging to σ1 , and sample a Markov chain beginning at vi 2 , and stopped at the first moment it hits some c G. R. Grimmett 6 February 2009

[2.2]

Wilson algorithm

21

vertex of σ1 . Call the resulting walk W2 , and loop-erase it to obtain some non-self-intersecting path LE(W2 ) from vi 2 to σ1 . Set σ2 = σ1 ∪ LE(W2 ), the union of σ1 and the directed path LE(W2 ). 4. Iterate the above process, by loop-erasing the trajectory of a Markov chain / σ j until it strikes the graph σ j previously starting at a new vertex vi j +1 ∈ constructed. 5. Stop when all vertices have been visited, and set σ = σ N , the final value of the σ j . (2.8) Theorem [213]. The graph σ is a spanning arborescence with root r , and P(σ = A) ∝ α( A),

A ∈ 6r .

Since S is finite and the chain is assumed irreducible, there exists a unique stationary distribution π = (πs : s ∈ S). Suppose that the chain is reversible with respect to π in that πx px,y = π y p y,x ,

x, y ∈ S.

As in Section 1.1, to each edge e = [x, yi we may allocate the weight w(e) = πx px,y , noting that the edges [x, yi and [y, xi have equal weight. Let A be a spanning arborescence with root r . Since each vertex of H other than the root is the head of a unique edge of the spanning arborescence A, we have by (2.7) that Q e∈A πe− pe− ,e+ α( A) = Q = C W ( A), A ∈ 6r , x∈S, x6=r πx where C = Cr and

W ( A) =

Y

w(e).

e∈A

Therefore, for a given root r , the weight functions α and W generate the same probability measure on 6r . The UST measure on G = (V, E) arises through a consideration of the random walk on G, i.e., by taking p x,y = 1/deg(x). By Theorem 2.8, Wilson’s algorithm generates a random spanning arborescence σ with given root on the graph H obtained from G by duplicating and directing the edges. When we neglect the orientations of the edges of σ , and also the identity of the root, σ is transformed into a uniform spanning tree of G. The remainder of this section is devoted to a proof of Theorem 2.8, and it uses the beautiful construction presented in [213]. For each x ∈ S \ {r }, we provide ourselves in advance with an infinite set of ‘moves’ from x. Let Mx (i ), i ≥ 1, x ∈ S \ {r }, be independent random variables with laws P(Mx (i ) = y) = px,y , y ∈ S. For each x, we organize the M x (i ) into an ordered ‘stack’. We think of an element Mx (i ) as having ‘colour’ i , where the colours indexed by i are distinct. The c G. R. Grimmett 6 February 2009

22

Uniform Spanning Tree

[2.2]

root r is given an empty stack. At stages of the following construction, we shall discard elements of stacks in order of increasing colour, and we shall call the set of uppermost elements of the stacks the ‘visible moves’. The visible moves generate a directed subgraph of H termed the ‘visible graph’. There will generally be directed cycles in the visible graph, and we shall remove such cycles one by one. Whenever we decide to remove a cycle, the corresponding visible moves are removed from the stacks, and a new set of moves beneath is revealed. The visible graph thus changes, and a second cycle may be removed. This process may be iterated until the earliest time, N say, at which the visible graph contains no cycle, which is to say that the visible graph is a spanning arborescence σ with root r . If N < ∞, we terminate the procedure and ‘output’ σ . The removal of a cycle is called ‘cycle popping’. It would seem that the value of N and the output σ will depend on the order in which we decide to pop cycles, but the converse turns out to be the case. The following lemma holds ‘pointwise’: it contains no statement involving probabilities. (2.9) Lemma. The order of cycle popping is irrelevant to the outcome, in that either: for all orderings of cycle popping, N = ∞,

or: the total number N of popped cycles, and the output σ , are independent of the order of popping.

Proof. A coloured cycle is a sequence M x j (i j ), j = 1, 2, . . . , J , that constitutes a cycle of H . A coloured cycle C is called poppable if there exists a sequence C 1 , C 2 , . . . , C n = C of coloured cycles that may be popped in order. We claim the following for any cycle-popping algorithm. If the algorithm terminates in finite time, then all poppable cycles are popped, and no others. The lemma follows from this claim. Let C be a poppable coloured cycle, and let C 1 , C 2 , . . . , C n = C be as above. It suffices to show the following. Let C 0 6= C 1 be a poppable cycle with colour 1, and suppose we pop C 0 at the first stage, rather than C 1 . Then C is still poppable after the removal of C 0 . Let V (D) denote the vertex-set of a coloured cycle D. The italicized claim is evident if V (C 0 ) ∩ V (C k ) = ∅ for k = 1, 2, . . . , n. Suppose on the contrary that V (C 0 ) ∩ V (C k ) 6= ∅ for some k, and let K be the earliest such k. Let x ∈ V (C 0 ) ∩ V (C K ). Since x ∈ / V (C k ) for k < K , the visible move at x has colour 1 even after the popping of C 1 , C 2 , . . . , C K −1 . Therefore, the edge of C K with head x has the same tail, y say, as that of C 0 with head x. This argument may be applied to y also, and then to all vertices of C K in order. In conclusion, C K has colour 1, and C 0 = C K .

Were we to decide to pop C 0 first, then we may choose to pop in the sequence C K [= C 0 ], C 1 , C 2 , C 3 , . . . , C K −1 , C K +1 , . . . , C n = C, and the claim has been shown.  c G. R. Grimmett 6 February 2009

[2.3]

Weak limits on lattices

23

Proof of Theorem 2.8. It is clear by construction that the Wilson algorithm terminates after finite time, with probability 1. It proceeds by popping cycles, and so, by Lemma 2.9, N < ∞ almost surely, and the output σ is independent of the choices available in its implementation. We show next that σ has the required law. We may think of the stacks as generating a pair (C, σ ), where C = (C 1 , C 2 , . . . , C J ) is the ordered set of coloured cycles that are popped by Wilson’s algorithm, and σ is the spanning arborescence thus revealed. Note that the colours of the moves of σ are determined by knowledge of C. Let C be the set of all sequences C that may occur, and 5 the set of all possible pairs (C, σ ). Certainly 5 = C × 6r , since knowledge of C imparts no information about σ . The law of (C, σ ) is simply the probability that the coloured moves are given appropriately. That is,


P (C, σ ) = (c, A) =

Y Y c∈c e∈c



pe− ,e+ α( A),

c ∈ C, A ∈ 6r .

Since this factorizes in the form f (c)g( A), the random variables C and σ are independent, and P(σ = A) is proportional to α( A) as required. 

2.3 Weak limits on lattices Let Ld = (Zd , Ed ) be the d-dimensional hypercubic lattice, with d ≥ 2. Let µn be the UST measure on the box B(n) = [−n, n]d . (2.10) Theorem [170]. The weak limit µ = lim n→∞ µn exists and is a translationinvariant and ergodic3 probability measure. It is supported on the set of forests in Ld with no bounded component. Since we are working in the σ -field of  generated by the cylinder events, it suffices for weak convergence4 that µn (B ⊆ T ) → µ(B ⊆ T ) for any finite set B of edges (see Exercise 2.3). Note that the limit measure µ may place strictly positive probability on the set of forests with two or more components. By a mild extension of the proof, one obtains that the limit measure µ is invariant under the action of any automorphism of the lattice Ld . Proof. Let F be a finite set of edges of Ed . By the Rayleigh principle, Theorem 1.29 (as in the proof of Theorem 2.1, see Exercise 2.4), µn (F ⊆ T ) ≥ µn+1 (F ⊆ T ), 3µ

4A

is ergodic if any shift-invariant event A has probability either 0 or 1. brief note about weak convergence can be found at the end of this section.

c G. R. Grimmett 6 February 2009

24

[2.3]

Uniform Spanning Tree

for all large n. Therefore, the limit µ(F ⊆ T ) = lim µn (F ⊆ T ) n→∞

exists. The domain of µ may be extended to all cylinder events, by the inclusion– exclusion principle, and this in turn specifies a unique probability measure µ on the infinite grid. Since no tree contains a cycle, and since each cycle is finite and there are countably many cycles in Ld , µ has support in the set of forests. By a similar argument, these forests may be taken with no bounded component. Let π be a translation of Z2 , and let F be finite as above. Then µ(π F ⊆ T ) = lim µn (π F ⊆ T ) = lim µπ,n (F ⊆ T ), n→∞

n→∞

where µπ,n is the law of a UST on π −1 B(n). There exists r = r (π ) such that B(n − r ) ⊆ π −1 B(n) ⊆ B(n + r ) for all large n. By the Rayleigh principle again, µn+r (F ⊆ T ) ≤ µπ,n (F ⊆ T ) ≤ µn−r (F ⊆ T ) for all large n. Therefore, µπ,n (F ⊆ T ) → µ(F ⊆ T ), whence the translation-invariance of µ. The proof of ergodicity is omitted, and may be found in [170].  This leads immediately to the question of whether or not the support of µ is the set of spanning trees of Ld . (2.11) Theorem [170]. The limit measure µ is supported on the set of spanning trees of Ld if and only if d ≤ 4.

The above measure µ may be termed ‘free UST measure’. There is another possible boundary condition giving rise to the so-called ‘wired UST measure’. One identifies as a single vertex all vertices not in B(n − 1), and chooses a spanning tree uniformly at random from the resulting (finite) graph. One can pass to the limit as n → ∞ in very much the same way as before. It turns out that the free and wired measures are identical on Ld for all d. The key fact is that Ld is a so-called amenable graph, which amounts in this context to saying that the boundary/volume approaches zero in the limit of large boxes, |∂ B(n)|/|B(n)| → 0

as n → ∞.

See Exercise 2.8 and [29, 157, 170, 171] for further details and discussion. This section closes with a brief note about weak convergence, for more details of which the reader is referred to the books [36, 67]. Let E = {ei : 1 ≤ i < ∞} be a countably infinite set. The product space  = {0, 1} E may be viewed as c G. R. Grimmett 6 February 2009

[2.4]

Uniform Forest

25

the product of copies of the discrete topological space {0, 1} and, as such,  is compact, and is metrisable by 0

δ(ω, ω ) =

∞ X i =1

2−i |ω(ei ) − ω0 (ei )|,

ω, ω0 ∈ .

A subset C of  is called a cylinder event (or, simply, a cylinder) if there exists a finite F ⊆ E such that: ω ∈ C if and only if ω 0 ∈ C for all ω0 equal to ω on F. The product σ -algebra F of  is the σ -algebra generated by the cylinders. The Borel σ -algebra B of  is defined as the minimal σ -algebra containing the open sets. It is standard that B is generated by the cylinders, and therefore F = B. We note that every cylinder is both open and closed in the product topology. Let (µn : n ≥ 1) and µ be probability measures on (, F ). We say that µn converges weakly to µ, written µn ⇒ µ, if µn ( f ) → µ( f )

as n → ∞,

for all bounded continuous functions f :  → R. (As usual, P( f ) denotes the expectation of the function f under the measure P.) Several other definitions of weak convergence are possible, and the so-called ‘portmanteau theorem’ asserts that certain of these are equivalent. In particular, the weak convergence of µ n to µ is equivalent to each of the two following statements: (i) lim supn→∞ µn (C) ≤ µ(C) for all closed events C, (ii) lim inf n→∞ µn (C) ≥ µ(C) for all open events C. The matter is simpler in the current setting: since the cylinder events are both open and closed, and they generate F , it is necessary and sufficient for weak convergence that (iii) limn→∞ µn (C) = µ(C) for all cylinders C. The following is useful for the construction of infinite-volume measures in the theory of interacting systems. Since  is compact, every family of probability measures on (, F ) is relatively compact. That is to say, for any such family 5 = (µi : i ∈ I ), every sequence (µn k : k ≥ 1) in 5 possesses a weakly convergent subsequence. Suppose now that (µn : n ≥ 1) is a sequence of probability measures on (, F ). If the limits limn→∞ µn (C) exists for every cylinder C, then it is necessarily the case that µ := limn→∞ µn exists and is a probability measure. We shall see in Exercises 2.2–2.3 that this holds if and only if lim n→∞ µn (C) exists for all increasing cylinders C. This justifies the argument of the proof of Theorem 2.10.

2.4 Uniform Forest We saw in Theorems 2.1 and 2.5 that the UST has a property of negative correlation. There is evidence that certain related measures have such a property also, but such claims have resisted proof. c G. R. Grimmett 6 February 2009

26

[2.5]

Uniform Spanning Tree

Let G = (V, E) be a finite graph, which we may as well assume to be connected. Write F for the set of forests of G (that is, subsets H ⊆ E containing no cycles), and C for the set of connected subgraphs of G (that is, subsets H ⊆ E such that (V, H ) is connected). Let F be a uniformly chosen member of F , and C a uniformly chosen member of C. We refer to F and C as a uniform forest (UF) and a uniform connected subgraph (USC), respectively. (2.12) Conjecture. For f, g ∈ E, f 6= g, the UF and USC satisfy: (2.13) (2.14)

P( f ∈ F | g ∈ F) ≤ P( f ∈ F), P( f ∈ C | g ∈ C) ≤ P( f ∈ C).

One may ask whether UF and USC satisfy the stronger conclusion of Theorem 2.5. As positive evidence of Conjecture 2.12, we cite the computer-aided proof of [111] that the UF on any graph with eight or fewer vertices (or nine vertices and eighteen or fewer edges) satisfies (2.13). Discuss general approaches to negative correlation, [131, 173].

2.5 Schramm–L¨owner evolutions There is a beautiful result of Lawler, Schramm, and Werner [147] concerning the limiting LERW (loop-erased random walk) and UST measures on L2 . This cannot be described without a detour into the theory of Schramm–Lo¨ wner evolutions5 (SLE). Let H = (−∞, ∞) × (0, ∞) be the upper half-plane of R2 , with closure H, viewed as subsets of the complex plane. Consider the (L o¨ wner) ordinary differential equation 2 d gt (z) = , dt gt (z) − b(t)

z ∈ H \ {0},

subject to the boundary condition g0 (z) = z, where t ∈ [0, ∞), and b : R → R is termed the ‘driving function’. Randomness in injected into this formula through setting b(t) = Bκt where κ > 0 and (Bt : t ≥ 0) is a standard Brownian motion6 . The solution exists when gt (z) is bounded away from Bκt . More specifically, for z ∈ H, let τz be the infimum of all times τ such that 0 is a limit point of gs (z) − Bκs in the limit as s ↑ τ . We let Ht = {z ∈ H : τz > t}, 5 Originally

K t = {z ∈ H : τz ≤ t},

known as ‘stochastic Lo¨ wner evolutions, but now often renamed after Schramm, in recognition of [189]. 6 See [68] for an interesting and topical account of the history and practice of Brownian motion. c G. R. Grimmett 6 February 2009

[2.5]

SLE2

SLE6

Schramm–L¨owner evolutions

0

SLE4

0

SLE8

27

0

0

Figure 2.1. Simulations of chordal SLEκ for κ = 2, 4, 6, 8. The four pictures are generated from the same Brownian driving path.

so that Ht is open, and K t is compact. It may now be seen that gt is a conformal homeomorphism from Ht to H. The process may be described via a random curve γ : [0, ∞) → H in the sense that H \ K t is the unbounded component of H \ γ [0, t]. The curve γ satisfies γ (0) = 0 and γ (t) → ∞ as t → ∞. See the illustrations of Figure 2.1. We call (gt : t ≥ 0) a Schramm–L¨owner evolution (SLE) with parameter κ, written SLEκ , and we call the K t the hulls of the process. There is good reason to believe that the family K = (K t : t ≥ 0) provides the correct scaling limits for a variety of random spatial processes, with the value of κ depending on the process in question. General properties of SLEκ , viewed as a function of κ, have been studied in [182, 208, 209], and a beautiful theory has emerged. For example, the hulls K form (almost surely) a simple path if and only if κ ≤ 4. If κ > 8, then SLEκ generates (almost surely) a space-filling curve. The above SLE is termed ‘chordal’. In another version, called ‘radial’ SLE, the upper half-plane H is replaced by the unit disc U, and a different differential equation is satisfied. The corresponding curve γ satisfies γ (t) → 0 as t → ∞, c G. R. Grimmett 6 February 2009

28

[2.5]

Uniform Spanning Tree

b

a Figure 2.2. The unique UST path between opposite corners a, b of a square. It has the law of a LERW between a and b.

and γ (0) ∈ ∂ U, say γ (0) = 1. Both chordal and radial SLE may be defined on an arbitrary simply connected domain D with a boundary, by applying a suitable conformal map φ from either H or U to D. Schramm [189, 190] identified the correct value of κ for several different processes, and indicated that percolation has scaling limit SLE6 . Full rigorous proofs are not yet known even for general percolation models. For the special case of site percolation on the triangular lattice T, Smirnov [196, 197] has proved the very remarkable result that the crossing probabilities of re-scaled regions of R 2 satisfy Cardy’s formula, see Section 5.6. The theory of SLE is a major piece of contemporary mathematics which promises to explain phase transitions in an important class of two-dimensional disordered systems, and to help bridge the gap between probability theory and conformal field theory. It has in addition provided complete explanations of conjectures made by mathematicians and physicists concerning the intersection exponents and fractionality of frontier of two-dimensional Brownian motion, see [144, 145]. This chapter closes with a brief summary of the results of [147] concerning SLE limits for LERW and UST on the square lattice L2 . We saw earlier in this chapter that there is a very close relationship between LERW and UST on a finite connected graph G. For example, the unique path joining vertices u and v in a UST of G has the law of a LERW from u to v (see [170] and the description of Wilson’s algorithm). See Figure 2.2. Let D be a bounded simply connected subset of C with 0 ∈ D. As remarked above, we may define radial SLE2 on D, and we write ν for its law. Let δ > 0, and let µδ be the law of LERW on the re-scaled lattice δ Z2 , starting at 0 and stopped c G. R. Grimmett 6 February 2009

[2.6]

Schramm–L¨owner evolutions

29

b dual UST UST Peano UST curve

a Figure 2.3. An illustration of the Peano UST path lying between a tree and its dual. The thinner continuous line depicts the UST, and the dashed line its dual tree. The thicker line is the Peano UST path.

when it first hits ∂ D. For two parametrizable curves β, γ in C, we define the distance between them by " # ρ(β, γ ) = inf

b −γ sup |β(t) b(t)| ,

t∈[0,1]

band γ where the infimum is over all parametrizations β b of the curves (see [8]). The distance function ρ generates a topology on the space of parametrizable curves, and hence a notion of weak convergence (denoted ‘⇒’). (2.15) Theorem [147]. We have that µδ ⇒ ν as δ → 0.

We turn to the convergence of UST to SLE8 , and begin with a discussion of mixed boundary conditions. Let D be a bounded simply connected domain of C with a smooth (C 1 ) boundary curve ∂ D. For distinct points a, b ∈ ∂ D, we write α (respectively, β) for the arc of ∂ D going clockwise from a to b (respectively, b to a). Let δ > 0 and let G δ be a connected graph that approximates to that part of δ Z2 lying inside D. We shall construct a UST of G δ with mixed boundary conditions, namely a free boundary near α and a wired boundary near β. To each tree T of G δ there corresponds a dual tree T d on the dual graph G dδ , namely the tree comprising edges of G dδ that do not intersect those of T . Since G δ has mixed boundary conditions, so does its dual G dδ . With G δ and G dδ drawn together, there is a simple path π(T , T d ) that winds between T and T d . Let 5 be the path thus constructed between the UST on G δ and its dual tree. The construction of this ‘Peano UST curve’ is illustrated in Figures 2.3 and 2.4. (2.16) Theorem [147]. The law of 5 converges as δ → 0 to that of the image of chordal SLE8 under any conformal map from H to D mapping 0 to a and ∞ to b.

c G. R. Grimmett 6 February 2009

30

[2.6]

Uniform Spanning Tree

Figure 2.4. An initial segment of the Peano path constructed from a UST on a large square with mixed boundary conditions.

2.6 Exercises 2.1. [17, 48] Aldous–Broder algorithm. Let G = (V, E) be a finite connected graph, and pick a root r ∈ V . Perform a random walk on G starting from r . For each v ∈ V , v 6= r , let ev be the edge traversedSby the random walk just before it hits v for the first time, and let T be the tree v ev rooted at r . Show that T , when viewed as an unrooted tree, is a uniform spanning tree. It may be helpful to argue as follows. a. Consider a stationary simple random walk (X n : −∞ < n < ∞) on G, with distribution πv ∝ deg(v), the degree of v. Let Ti be the rooted tree obtained by the above procedure applied to the sub-walk X i , X i +1 , . . . . Show that T = (Ti : −∞ < i < ∞) is a stationary Markov chain with state space the set R of rooted spanning trees. b. Let Q(t, t 0 ) = P(T0 = t 0 | T1 = t), and let r (t) be the degree of the root of t ∈ R. Show that: (i) for given t ∈ R, there are exactly r (t) trees t 0 ∈ R with Q(t, t 0 ) = 1/r (t), and Q(t, t 0 ) = 0 for all other t 0 , (ii) for given t 0 ∈ R, there are exactly r (t 0 ) trees t ∈ R with Q(t, t 0 ) = 1/r (t), and Q(t, t 0 ) = 0 for all other t. c. Show that X r (t)Q(t, t 0 ) = r (t 0 ), t 0 ∈ R, t∈R

and deduce that the stationary measure of T is proportional to r (t). d. Let r ∈ V , and let t be a tree with root r . Show that P(T0 = t | X 0 = r ) is independent of the choice of t. 2.2. Let  = {0, 1} F where F is finite, and let P be a probability measure on c G. R. Grimmett 6 February 2009

[2.6]

Exercises

31

, and A ⊆ . Show that P( A) may be expressed as a linear combination of certain P( Ai ) where the Ai are increasing events. 2.3. (continuation) Let G = (V, E) be an infinite graph with finite vertexdegrees, and  = {0, 1} E . An event A in the product σ -field of  is called a cylinder event if it has the form A F × {0, 1} F for some A F ⊆ {0, 1} F and some finite F ⊆ E. Show that a sequence (µn ) of probability measures converges weakly if µn ( A) converges for every increasing cylinder event A. 2.4. Let G = (V, E) be a finite connected subgraph of the finite connected graph G 0 . Let T and T 0 be uniform spanning trees on G and G 0 respectively. Show that, for any edge e of G, P(e ∈ T ) ≥ P(e ∈ T 0 ). More generally, let B be a subset of E, and show that P(B ⊆ T ) ≥ P(B ⊆ T 0 ). 2.5. Let Tn be a UST of the lattice box [−n, n]d of Zd . Show that the limit λ(e) = limn→∞ P(e ∈ Tn ) exists. More generally, show that the weak limit of Tn exists as n → ∞. 2.6. Adapt the conclusions of the last two examples to the ‘wired’ UST measure µw on Ld . 2.7. Let F be the set of forests of Ld with no bounded component, and let µ be an automorphism-invariant probability measure with support F . Show that the mean degree of every vertex is 2. 2.8. [170] Let A be an increasing cylinder event in {0, 1}E . Using the Feder– Mihail Theorem 2.5 or otherwise, show that the free and wired UST measures on Ld satisfy µf ( A) ≥ µw ( A). Deduce by the last exercise and Strassen’s theorem, or otherwise, that µf = µw . 2.9. Consider the square lattice L2 as an infinite electrical network with unit edge-resistances. Show that the effective resistance between two neighbouring vertices is 2. 2.10. Let G = (V, E) be finite and connected, and let W ⊆ V . Let F W be the set of forests of G comprising exactly |W | trees with respective roots the members of W . Explain how Wilson’s algorithm may be adapted to sample uniformly from FW . d

c G. R. Grimmett 6 February 2009

3 Percolation and Self-Avoiding Walk

The central feature of the percolation model is the phase transition. The existence of the point of transition is proved by path-counting and planar duality. Basic facts about self-avoiding walks, oriented percolation, and the coupling of models are reviewed.

3.1 Phase transition Percolation is the fundamental stochastic model for spatial disorder. In its simplest form introduced in [47]1 , it inhabits a (crystalline) lattice and possesses the maximum of (statistical) independence. We shall consider mostly percolation on the (hyper)cubic lattice Ld = (Zd , Ed ) in d ≥ 2 dimensions, but much of the following may be adapted to an arbitrary lattice. Percolation comes in two forms, ‘bond’ and ‘site’, and we concentrate here on the bond model. Let p ∈ [0, 1]. Each edge e ∈ Ed is designated either open with probability p, or closed otherwise, different edges receiving independent designations. We think of an open edge as being open to the passage of some material such as disease, liquid, or infection; closed edges are closed to such passage. Suppose we remove all closed edges, and consider the remaining open subgraph of the lattice. Percolation theory is concerned with the geometry of this open graph. Of particular interest are such quantites as the size of the open cluster C x containing a given vertex x, and particularly the probability that C x is infinite. The sample space is the set  = {0, 1}E of 0/1-vectors ω indexed by the edgeset; here, 1 represents ‘open’, and 0 ‘closed’. As σ -field we take that generated by the finite-dimensional cylinder sets, and the relevant probability measure is product measure P p with density p. d

For x, y ∈ Zd , we write x ↔ y if there exists an open path joining x and y. The open cluster C x at x is the set of all vertices reachable along open paths from 1 See

also [214].

c G. R. Grimmett 6 February 2009

[3.1]

the vertex x:

Phase transition

33

C x = {y ∈ Zd : x ↔ y}.

The origin of Zd is denoted 0, and we write C = C 0 . The principal object of study is the percolation probability θ ( p) given by θ ( p) = P p (|C| = ∞). The critical probability is defined as (3.1)

pc = pc (Ld ) = sup{ p : θ ( p) = 0}.

It is fairly clear (and we will spell this out soon) that θ is non-decreasing in p, and thus  = 0 if p < pc , θ ( p) > 0 if p > pc .

It is fundamental that 0 < pc < 1, and we state this as a theorem. It is easy to see that pc = 1 for the corresponding one-dimensional process. (3.2) Theorem. For d ≥ 2, we have that 0 < pc < 1.

It is an important open problem to prove the following conjecture. The conclusion is known only for d = 2 and d ≥ 19. (3.3) Conjecture. For d ≥ 2, we have that θ ( pc ) = 0.

It is the edges (or ‘bonds’) of the lattice that are declared open/closed above. If, instead, we designate the vertices (or ‘sites’) to be open/closed, the ensuing model is termed site percolation. Subject to minor changes, the theory of site percolation may be developed just as that of bond percolation. Proof of Theorem 3.2. This proof introduces two basic methods, namely the counting of paths and the use of planar duality. We show first by counting paths that pc > 0. A self-avoiding walk (SAW) is a lattice path that visits no vertex more than once. Let σn be the number of SAWs with length n beginning at the origin, and let Nn be the number of such SAWs all of whose edges are open. Then θ ( p) = P p (Nn ≥ 1 for all n ≥ 1) = lim P p (Nn ≥ 1). n→∞

Now, (3.4)

P p (Nn ≥ 1) ≤ P p (Nn ) = p n σn .

A a crude upper bound for σn , we have that (3.5)

σn ≤ 2d(2d − 1)n−1 ,

c G. R. Grimmett 6 February 2009

n ≥ 1,

34

[3.1]

Percolation and Self-Avoiding Walk

Figure 3.1. Part of the square lattice 2 and its dual.

since the first step of a SAW from the origin can be to any of its 2d neighbours, and there are no more than 2d − 1 choices for each subsequent step. Thus θ ( p) ≤ lim 2d(2d − 1)n−1 p n , n→∞

which equals 0 whenever p(2d − 1) < 1. Therefore, pc ≥

1 . 2d − 1

We turn now to the proof that pc < 1. The first step is to observe that (3.6)

pc (Ld ) ≥ pc (Ld+1 ),

d ≥ 2.

This follows immediately by the observation that Ld may be embedded in Ld+1 in such a way that the origin lies in an infinite open cluster of Ld+1 whenever it lies in an infinite open cluster of the smaller lattice Ld . By (3.6), it suffices to show that (3.7)

pc (L2 ) < 1,

and to this end we shall use planar duality. The square lattice has a special property, namely that of self-duality. Planar duality arises as follows. Let G be a planar graph, drawn in the plane. The planar dual of G is the graph constructed in the following way. We place a vertex in every face of G (including the infinite face if it exists) and we join two such vertices by an edge if and only if the corresponding faces of G share a boundary edge. It is easy to see that the dual of the square lattice c G. R. Grimmett 6 February 2009

[3.2]

Phase transition

35

Figure 3.2. A finite open cluster of the primal lattice lies ‘just inside’ an closed cycle of the dual lattice.

L2 is a copy of L2 , and we refer therefore to the square lattice as being self-dual.

See Figure 3.1. There is a natural one–one correspondence between the edge-set of the dual lattice L2d and that of the primal L2 , and this gives rise to a percolation model on L2d by: for an edge e ∈ E2 and it dual edge ed , we declare ed to be open if and only if e is open. As illustrated in Figure 3.2, each finite open cluster of L 2 lies in the interior of a closed cycle of L2d . We use a ‘Peierls argument’ to obtain (3.7)2 . Let Mn be the number of closed circuits of the dual lattice, having length n and containing 0 in their interior. Note that |C| < ∞ if and only if Mn ≥ 1 for some n. Therefore,

(3.8)

1 − θ ( p) = P p (|C| < ∞) = P p ≤ Pp =

X

∞ X n=4

n

Mn



P p (Mn ) ≤

∞ X n=4

X n

Mn ≥ 1



(n4n )(1 − p)n ,

where we have used the facts that the shortest dual circuit containing 0 has length 4, and that the total number of dual circuits, having length n and surrounding the origin, is no greater than n4n . The final sum may be made strictly smaller than 1 by choosing p sufficiently close to 1, say p > 1 −  where  > 0. This implies that pc (L2 ) < 1 −  as required for (3.7). 

2 This

method was used by Peierls [169] to prove phase transition for the two-dimensional Ising model. c G. R. Grimmett 6 February 2009

36

[3.2]

Percolation and Self-Avoiding Walk

3.2 Self-avoiding walks How many self-avoiding walks of length n exist, starting from the origin? What is the ‘shape’ of a SAW chosen at random from this set? In particular, what is the distance between its endpoints? These and related questions have attracted a great deal of attention since the publication in 1954 of the pioneering paper [116] of Hammersley and Morton, and never more so than in recent years. It is believed but not proved that a typical SAW on L2 , starting at the origin, converges in a suitable manner as n → ∞ to a SLE8/3 curve, and the proof of this statement is an open problem of outstanding interest. See Section 2.5, in particular Figure 2.1, and [159, 190, 198]. The use of subadditivity was one of the several stimulating ideas of [116], and it has proved extremely fruitful in many contexts since. Consider the lattice L d , and let Sn be the set of SAWs with length n starting at the origin, and σn = |Sn | as before. (3.9) Lemma. We have that σm+n ≤ σm σn , for m, n ≥ 0. Proof. Let π and π 0 be finite SAWs starting at the origin, and denote by π ∗ π 0 the walk obtained by following π from 0 to its other endpoint x, and then following the translated walk π 0 + x. Every ν ∈ Sm+n may be written in a unique way as ν = π ∗ π 0 for some π ∈ Sm and π 0 ∈ Sn . The claim of the lemma follows.  (3.10) Theorem [116]. The limit κ = limn→∞ (σn )1/n exists and satisfies d ≤ κ ≤ 2d − 1.

This is in essence a consequence of the ‘sub-multiplicative’ inequality of Lemma 3.9, see Exercise 3.1. The constant κ is called the connective constant of the lattice. The exact value of κ = κ(Ld ) is unknown for every d ≥ 2, see Section 7.2 of [126], pp. 481–483. It is conjectured that the connective constant p of√the ‘hexagonal’ (or ‘honeycomb’) lattice, illustrated in Figure 5.9, equals 2 + 2.

Proof. By Lemma 3.9, x m = log σm satisfies the ‘subadditive inequality’ (3.11)

x m+n ≤ x m + x n .

The existence of the limit λ = lim

n→∞



1 xn n



follows immediately, and λ = inf m {x m /m} ∈ [−∞, ∞). By (3.5), κ = e λ ≤ 2d − 1. Finally, σn is at least the number of ‘stiff’ walks every step of which is in the direction of an increasing coordinate. The number of such walks is d n , and therefore κ ≥ d.  The bounds of Theorem 3.2 may be improved as follows. c G. R. Grimmett 6 February 2009

[3.3]

Self-avoiding walks

37

(3.12) Theorem. The critical probability of bond percolation on L d , with d ≥ 2, satisfies 1 1 ≤ pc ≤ 1 − , κ(d) κ(2) where κ(d) denotes the connective constant of Ld . Proof. As in (3.4),

θ ( p) ≤ lim p n σn . n→∞

Now, σn = κ(d)(1+o(1))n , so that θ ( p) = 0 if pκ(d) < 1. For the upper bound, we elaborate on the proof of the corresponding part of Theorem 3.2. Let Fm be the event that there exists a closed cycle of the dual lattice L2d containing the primal box B(m) = [−m, m]2 in its interior, and let G m be the event that all edges of B(m) are open. These two events are independent, since they are defined in terms of disjoint sets of edges. As in (3.8), (3.13)

P p (Fm ) ≤ P p

≤

X ∞

∞ X

n=4m

n=4m

Mn ≥ 1



n(1 − p)n σn .

Recall that σn = κ(2)(1+o(1))n , and choose p such that (1 − p)κ(2) < 1. By (3.13), we may find m such that P p (Fm ) < 12 . Then, θ ( p) ≥ P p (Fm ∩ G m ) = P p (Fm )P p (G m ) ≥ 12 P p (G m ) > 0. The upper bound on pc follows.



There are some extraordinary conjectures concerning SAWs in two dimensions. We mention the conjecture that σn ∼ An 11/32 κ n

when d = 2,

expected to hold for any lattice in two dimensions, with an appropriate choice of constant A depending on the choice of lattice. It is known in contrast that no polynomial correction is necessary when d ≥ 5, σn ∼ Aκ n

when d ≥ 5,

for the cubic lattice at least. See [159, 190, 198] for further details of these and other conjectures and results.

c G. R. Grimmett 6 February 2009

38

[3.4]

Percolation and Self-Avoiding Walk

3.3 Coupled percolation The use of coupling in probability theory goes back at least as far as the beautiful proof by Doeblin of the ergodic theorem for Markov chains, [65]. In percolation, we couple together the bond models with different values of p as follows. Let U e , e ∈ Ed , be independent random variables with the uniform distribution on [0, 1]. For p ∈ [0, 1], let  1 if Ue < p, η p (e) = 0 otherwise.

Thus the configuration η p (∈ ) has law P p , and in addition η p ≤ ηr

if

p ≤ r.

(3.14) Theorem. For any increasing non-negative random variable f :  → , the function g( p) = P p ( f ) is non-decreasing.

Proof. For p ≤ r ,

g( p) = P( f (η p )) ≤ P( f (ηr )) = g(r ), as required, where P denotes ‘generic probability’.



3.4 Oriented percolation E d obtained by orienting each edge of Ld in the direction The ‘north–east’ lattice L of increasing coordinate-value (see Figure 3.3 for a two-dimensional illustration). There are many parallels between results for oriented percolation and those for ordinary percolation; on the other hand the corresponding proofs often differ, largely because the existence of one-way streets restricts the degree of spatial freedom of the traffic. E d to be open with probability p and Let p ∈ [0, 1]. We declare an edge of L otherwise closed. The states of different edges are taken to be independent. We now supply fluid at the origin, and allow it to travel along open edges in the directions of their orientations only. Let CE be the set of vertices that may be reached from the origin along open directed paths. The percolation probability is (3.15)

E = ∞), E p) = P p (|C| θ(

and the critical probability pEc (d) by (3.16)

pEc (d) = sup{ p : θE( p) = 0}.

c G. R. Grimmett 6 February 2009

[3.4]

Oriented percolation

39

Figure 3.3. Part of the two-dimensional ‘north–east’ lattice in which each edge has been deleted with probability 1 − p, independently of all other edges.

(3.17) Theorem. For d ≥ 2, we have that 0 < pEc (d) < 1. Proof. Since an oriented path is also a path, it is immediate that θE( p) ≤ θ ( p), whence pEc (d) ≥ pc . As in the proof of Theorem 3.2, it suffices for the converse to show that pEc = pEc (2) < 1. Let d = 2. The cluster CE comprises the endvertices of open edges that are oriented northwards/eastwards. Assume |CE | < ∞. Surrounding CE one may draw a cycle 1 of the dual in the manner illustrated in Figure 3.4. As we traverse 1 in the clockwise direction, we traverse edges each of which is oriented in one of the four compass directions. Any edge of 1 that is oriented either eastwards or southwards crosses a primal edge that is closed. Exactly one half of the edges of 1 are oriented thus, so that, as in (3.8), E < ∞) ≤ P p (|C|

X n≥4

4 · 3n−2 (1 − p) 2 n−1 . 1

In particular, θE( p) > 0 if 1 − p is sufficiently small and positive.



The process is understood quite well when d = 2, see [70]. By looking at E 2, the set An of wet vertices on the diagonal {x ∈ Z2 : x 1 + x 2 = n} of L one may reformulate two-dimensional oriented percolation as a one-dimensional contact process in discrete time (see [148], Chapter 6)). It turns out that pEc (2) may be characterized in terms of the velocity of the rightwards edge of a contact process on Z whose initial distribution places infectives to the left of the origin and c G. R. Grimmett 6 February 2009

40

Percolation and Self-Avoiding Walk

[3.5]

1

Figure 3.4. As we trace the dual cycle 1, we traverse edges exactly one half of which cross E at the origin. closed boundary edges of the cluster C

susceptibles to the right. With the support of arguments from branching processes and ordinary percolation, one may prove such results as the exponential decay of the cluster size distribution when p < pEc (2), and its sub-exponential decay when p > pEc (2): there exist α( p), β( p) > 0 such that (3.18)

e−α( p)

√

n

√
≤ P p n ≤ |CE | < ∞ ≤ e−β( p) n

if pEc (2) < p < 1.

There is a close relationship between oriented percolation and the contact model (see Chapter 6), and methods developed for the latter model may often be applied to the former. It has been shown in particular that θE( pEc ) = 0 for general d ≥ 2, see [100]. We close this section with an open problem of a different sort. Suppose that each edge of L2 is oriented in a random direction, horizontal edges being oriented eastwards with probability p and westwards otherwise, and vertical edges being oriented northwards with probability p and southwards otherwise. Let η( p) be the probability that there exists an infinite oriented path starting at the origin. It is not hard to show that η( 12 ) = 0 (see Exercise 3.9. We ask whether or not η( p) > 0 if p 6= 12 . Partial results in this direction may be found in [97].

3.5 Exercises

c G. R. Grimmett 6 February 2009

[3.5]

Exercises

41

3.1. Subadditive inequality. Let (x n : n ≥ 1) be a real sequence satisfying x m+n ≤ x m + x n for m, n ≥ 1. Show that the limit λ = limn→∞ {x n /n} exists and satisfies λ = inf k {x k /k}. 3.2. (continuation) Find reasonable conditions on the sequence (α n ) such that: the generalized inequality x m+n ≤ x m + x n + αm ,

m, n ≥ 1,

implies the existence of the limit λ = lim n→∞ {x n /n}? 3.3. [108] Bond/site critical probabilities. Let G be an infinite connected graph with maximal vertex degree 1. Show that the critical probabilities for bond and site percolation on G satisfy pcbond ≤ pcsite ≤ 1 − [1 − pcbond ]1 . The second inequality is in fact valid with 1 replaced by 1 − 1. 3.4. Show that bond percolation on a graph G may be reformulated in terms of site percolation on a graph derived suitably from G. 3.5. Show that the connective constant of L2 lies strictly between 2 and 3. 3.6. One-dimensional percolation. Each edge of the one-dimensional lattice Z is declared open with probability p. For k ∈ Z, let r (k) = max{u : k ↔ k + u}, and L n = max{r (k) : 1 ≤ k ≤ n}. Show that P p (L n > u) ≤ np u , and deduce that, for  > 0,   (1 + ) log n Pp L n > →0 as n → ∞. log(1/ p) [This is the famous problem of the longest run of heads in n tosses of a coin.] 3.7. (continuation) Show that, for  > 0,   (1 − ) log n Pp L n < →0 as n → ∞. log(1/ p) By suitable refinements of the error estimates above, show that   (1 + ) log n (1 − ) log n < Ln < , for all but finitely many n = 1. Pp log(1/ p) log(1/ p) 3.8. Show the strict inequality pc (d) < pEc (d) for the critical probabilities of unoriented and oriented percolation on Ld with d ≥ 2. 3.9. [97] Each edge of the square lattice L2 is oriented in a random direction, horizontal edges being oriented eastwards with probability p and westwards otherwise, and vertical edges being oriented northwards with probability p and southwards otherwise. Let η( p) be the probability that there exists an infinite oriented path starting at the origin. Show that η( 12 ) = 0. c G. R. Grimmett 6 February 2009

42

Percolation and Self-Avoiding Walk

[3.5]

3.10. The vertex (i, j ) of L2 is called even if i + j is even, and odd otherwise. Vertical edges are oriented from the even endpoint to the odd, and horizontal edges vice versa. Each edge is declared open with probability p, and closed otherwise (independently between edges). Show that, for p sufficiently close to 1, there is strictly positive probability that the origin is the endpoint of an infinite open oriented path. 3.11. A word is an element of the set {0, 1}N of singly-infinite 0/1 sequences. Let p ∈ (0, 1) and M ≥ 1. Consider oriented site percolation on Z2 , in which the colour ω(x) of a vertex x equals 1 with probability p, and 0 otherwise. A word w = (w1 , w2 , . . . ) is said to be M-seen if there exists an infinite oriented path x 0 = 0, x 1 , x 2 , . . . of vertices such that ω(x i ) = wi and d(xi −1 , xi ) ≤ M for i ≥ 1. [Here, as usual, d denotes graph-theoretic distance.] Calculate the probability that the square {1, 2, . . . , k}2 contains both a 0 and a 1. Deduce3 by a block argument that ψ p (M) = Pp (all words are M-seen) satisfies ψ p (M) > 0 for M ≥ M( p), and determine an upper bound on the required M( p).

3 This

provides a short proof of the main result of [152].

c G. R. Grimmett 6 February 2009

4 Correlation and Concentration

Correlation-type inequalities have played a significant role in the theory of disordered spatial systems. The Holley inequality provides a sufficient condition for the stochastic ordering of two measures, and also a route to a proof of the famous FKG inequality. For product measures, the complementary BK inequality involves the concept of ‘disjoint occurrence’. Two concepts of concentration are considered here. The Hoeffding inequality provides a bound on the tail of a martingale with bounded differences. Another concept of ‘influence’ proved by Kahn, Kalai, and Linial leads to sharp-threshold theorems for increasing events under either product or FKG measures.

4.1 Holley inequality We review the stochastic ordering of probability measures on a discrete space. Let E be a non-empty finite set, and  = {0, 1} E . The sample space  is partially ordered by: ω1 ≤ ω2 if ω1 (e) ≤ ω2 (e) for all e ∈ E. A non-empty subset A ⊆  is called increasing if: ω ∈ A, ω ≤ ω0

⇒

ω0 ∈ A.

The subset A is decreasing if its complement A =  \ A is increasing.

(4.1) Definition. Given two probability measures µi , i = 1, 2, on , we write µ1 ≤st µ2 if µ1 ( A) ≤ µ2 ( A) for all increasing events A.

Equivalently, µ1 ≤st µ2 if and only if µ1 ( f ) ≤ µ2 ( f ) for all increasing functions f :  → R. There is an important and useful result, often termed Strassen’s theorem, that asserts that measures satisfying µ1 ≤st µ2 may be coupled in a ‘pointwise monotone’ manner. Such a statement is valid for very general spaces (see [153]), but we restrict ourselves here to the current context. The proof is omitted, and may be found in many places including [157]. c G. R. Grimmett 6 February 2009

44

[4.1]

Correlation and Concentration

(4.2) Theorem [201]. Let µ1 and µ2 be probability measures on . The following two statements are equivalent. (i) µ1 ≤st µ2 . (ii) There exists a probability measure ν on 2 such that ν({(π, ω) : π ≤ ω}) = 1, and whose marginal measures are µ1 and µ2 . For ω1 , ω2 ∈ , we define the (pointwise) maximum and minimum configurations by ω1 ∨ ω2 (e) = max{ω1 (e), ω2 (e)},

(4.3)

ω1 ∧ ω2 (e) = min{ω1 (e), ω2 (e)},

for e ∈ E. A probability measure µ on  is called positive if µ(ω) > 0 for all ω ∈ . (4.4) Holley inequality [125]. Let µ1 and µ2 be positive probability measures on  satisfying (4.5)

µ2 (ω1 ∨ ω2 )µ1 (ω1 ∧ ω2 ) ≥ µ1 (ω1 )µ2 (ω2 ),

ω1 , ω2 ∈ .

Then µ1 ≤st µ2 .

Proof. The main step is the proof that µ1 and µ2 can be ‘coupled’ in such a way that the component with marginal measure µ2 lies above (in the sense of sample realizations) that with marginal measure µ1 . This is achieved by constructing a certain Markov chain with the coupled measure as unique invariant measure. Here is a preliminary calculation. Let µ be a positive probability measure on . We can construct a time-reversible Markov chain with state space  and unique invariant measure µ by choosing a suitable generator G satisfying the detailed balance equations. The dynamics of the chain involve the ‘switching on or off’ of components of the current state. For ω ∈  and e ∈ E, we define the configurations ω e , ωe by   ω( f ) if f 6= e, ω( f ) if f 6= e, e (4.6) ω (f) = ωe ( f ) = 1 if f = e, 0 if f = e. Let G : 2E → R be given by (4.7)

G(ωe , ωe ) = 1,

G(ωe , ωe ) =

µ(ωe ) , µ(ωe )

for all ω ∈ , e ∈ E. Set G(ω, ω 0 ) = 0 for all other pairs ω, ω 0 with ω 6= ω0 . The diagonal elements are chosen in such a way that X G(ω, ω0 ) = 0, ω ∈ . ω0 ∈

c G. R. Grimmett 6 February 2009

[4.1]

Holley inequality

45

It is elementary that µ(ω)G(ω, ω0 ) = µ(ω0 )G(ω0 , ω),

ω, ω0 ∈ ,

and therefore G generates a time-reversible Markov chain on the state space . This chain is irreducible (using (4.7)), and therefore possesses a unique invariant measure µ (see [109], Theorem 6.5.4). We next follow a similar route for pairs of configurations. Let µ1 and µ2 satisfy the hypotheses of the theorem, and let S be the set of all pairs (π, ω) of configurations in  satisfying π ≤ ω. We define H : S × S → R by

(4.8)

H (πe , ω; π e , ωe ) = 1,

µ2 (ωe ) , µ2 (ωe ) µ1 (πe ) µ2 (ωe ) (4.10) H (π e , ωe ; πe , ωe ) = − , µ1 (π e ) µ2 (ωe ) for all (π, ω) ∈ S and e ∈ E; all other off-diagonal values of H are set to 0. The diagonal terms are chosen in such a way that X H (π, ω; π 0 , ω0 ) = 0, (π, ω) ∈ S. (4.9)

H (π, ωe ; πe , ωe ) =

π 0 ,ω0

Equation (4.8) specifies that, for π ∈  and e ∈ E, the edge e is acquired by π (if it does not already contain it) at rate 1; any edge so acquired is added also to ω if it does not already contain it. (Here, we speak of a configuration ψ containing an edge e if ψ(e) = 1.) Equation (4.9) specifies that, for ω ∈  and e ∈ E with ω(e) = 1, the edge e is removed from ω (and also from π if π(e) = 1) at the rate given in (4.9). For e with π(e) = 1, there is an additional rate given in (4.10) at which e is removed from π but not from ω. We need to check that this additional rate is indeed non-negative, and the required inequality, µ2 (ωe )µ1 (πe ) ≥ µ1 (π e )µ2 (ωe ),

π ≤ω

follows from assumption (4.5). Let (X t , Yt )t≥0 be a Markov chain on S with generator H , and set (X 0 , Y0 ) = (0, 1), where 0 (respectively, 1) is the state of all 0’s (respectively, 1’s). By examination of (4.8)–(4.10) we see that X = (X t )t≥0 is a Markov chain with generator given by (4.7) with µ = µ1 , and that Y = (Yt )t≥0 arises similarly with µ = µ2 . Let κ be an invariant measure for the paired chain (X t , Yt )t≥0 . Since X and Y have (respective) unique invariant measures µ1 and µ2 , the marginals of κ are µ1 and µ2 . We have by construction that κ(S) = 1, and κ is the required ‘coupling’ of µ1 and µ2 . Let (π, ω) ∈ S be chosen according to the measure κ. Then µ1 ( f ) = κ( f (ω)) ≤ κ( f (π )) = µ2 ( f ),

for any increasing function f . Therefore µ1 ≤st µ2 . c G. R. Grimmett 6 February 2009



46

[4.3]

Correlation and Concentration

4.2 FKG inequality The FKG inequality for product measures was discovered by Harris [121], and is often named now after the authors of [83] who proved the more general version that is the subject of this section. See the appendix of [98] for a historical account. Let E be a finite set, and  = {0, 1} E as usual. (4.11) Theorem. FKG inequality [83]. Let µ be a positive probability measure on  such that (4.12)

µ(ω1 ∨ ω2 )µ(ω1 ∧ ω2 ) ≥ µ(ω1 )µ(ω2 ),

ω1 , ω2 ∈ .

Then µ is ‘positively associated’ in that (4.13)

µ( f g) ≥ µ( f )µ(g)

for all increasing random variables f, g :  → R. It is explained in [83] how the condition of (strict) positivity can be removed. Condition (4.12) is called the ‘FKG lattice condition’. Proof. Assume that µ satisfies (4.12), and let f and g be increasing functions. By adding a constant to the function g, we see that it suffices to prove (4.13) under the additional hypothesis that g is strictly positive. We assume this holds. Define positive probability measures µ1 and µ2 on  by µ1 = µ and µ2 (ω) = P

g(ω)µ(ω) , 0 0 ω0 g(ω )µ(ω )

ω ∈ .

Since g is increasing, (4.5) follows from (4.12). By the Holley inequality, Theorem 4.4, µ1 ( f ) ≤ µ2 ( f ), which is to say that

as required.

P X ω f (ω)g(ω)µ(ω) P ≥ f (ω)µ(ω) 0 0 ω0 g(ω )µ(ω ) ω

c G. R. Grimmett 6 February 2009



[4.3]

BK inequality

47

4.3 BK inequality In the special case of product measure on , there is a type of converse inequality to the FKG inequality, named the BK inequality after the authors of [32]. This is based on a concept of ‘disjoint occurrence’ that we make more precise as follows. For ω ∈  and F ⊆ E we define the cylinder event C(ω, F) generated by ω on F by C(ω, F) = {ω0 ∈  : ω0 (e) = ω(e) for all e ∈ F} = (ω(e) : e ∈ F) × {0, 1} E \F .

We define the event A  B as the set of all ω ∈  for which there exists a set F ⊆ E such that C(ω, F) ⊆ A and C(ω, E \ F) ⊆ B. Thus, A  B is the set of configurations ω for which there exist disjoint sets F, G of indices with the property that: knowledge of ω restricted to F (respectively, G) implies that ω ∈ A (respectively, ω ∈ B). In the special case when A and B are increasing, C(ω, F) ⊆ A if and only if ω F ∈ A, where  ω(e) for e ∈ F, ω F (e) = 0 for e ∈ / F. Thus, in this case, A  B = A ◦ B where  A ◦ B = ω : there exists F ⊆ E such that ω F ∈ A, ω E \F ∈ B .

The set F is permitted to depend on the choice of configuration ω. Note that A  B ⊆ A ∩ B. Furthermore, if A and B are increasing, then so is A  B (= A ◦ B). Let P be the product measure on  with local densities pe , e ∈ E, that is Y P= µe , e∈E

where µe (0) = 1 − pe and µe (1) = pe .

(4.14) Theorem. BK inequality [32]. For increasing subsets A, B of , (4.15)

P( A ◦ B) ≤ P( A)P(B).

It is not known for what non-product measures (4.15) holds. It seems reasonable, for example, to conjecture that (4.15) holds for the measure Pk that selects a k-subset of E uniformly at random. It would be very useful to show that the random-cluster measure φ p,q on  satisfies (4.15) whenever 0 < q < 1, although we may have to survive with rather less. See Chapter 8, and Section 3.9 of [98]. The conclusion of the BK inequality is in fact valid for all pairs A, B of events, regardless of whether or not they are increasing. This is much harder to prove, and has not yet been as valuable as originally expected in the analysis of disordered systems. c G. R. Grimmett 6 February 2009

48

[4.3]

Correlation and Concentration

(4.16) Theorem. Reimer inequality [181]. For A, B ⊆ , P( A  B) ≤ P( A)P(B). One can see that A  B = A ∩ B if A is increasing and B is decreasing. By applying Reimer’s inequality to the events A and B, where A and B are increasing, we obtain that P( A ∩ B) ≥ P( A)P(B). Therefore, Reimer’s inequality includes both the FKG and BK inequalities for the product measure P. Proof of Theorem 4.14. We present the ‘simple’ proof of [30], see also [95, 210]. Those who prefer proofs by induction are directed to [44]. Let 1, 2, . . . , N be an ordering of E. We shall consider the duplicated sample space  ×  0 where  = 0 = {0, 1} E , with which we associate the product measure P = P × P. Elements of  (respectively, 0 ) are written as ω (respectively, ω 0 ). Let A and B be increasing subsets of {0, 1} E . For j ≥ 1 and (ω, ω0 ) ∈  × 0 , define the N -vector ω j by
ω j = ω0 (1), ω0 (2), . . . , ω0 ( j − 1), ω( j ), . . . , ω(N ) , bj , b and the events A B of  × 0 by

bj = {(ω, ω0 ) : ω j ∈ A}, A

b B = {(ω, ω0 ) : ω ∈ B}.

Note that: b1 = A × 0 and b bA b1 ◦ b (a) A B = B × 0 , so that P( B) = P( A ◦ B), bN +1 and b (b) A B are defined in terms of disjoint subsets of E, so that bA bN +1 ◦ b bA bN +1 ) P( bb P( B) = P( B) = P( A)P(B).

It thus suffices to show that (4.17)

bA bj ◦ b bA bj +1 ◦ b P( B) ≤ P( B),

1 ≤ j ≤ N,

and this we do, for given j , by conditioning on the values of the ω(i ), ω 0 (i ) for all i 6= j . Suppose these values are given, and classify them as follows. There are three cases. bj ◦ b 1. A B does not occur when ω( j ) = ω 0 ( j ) = 1. bj ◦ b bj +1 ◦ b 2. A B occurs when ω( j ) = ω0 ( j ) = 0, in which case A B occurs also. 3. Neither of the two cases above hold. bj ◦ b Consider the third case. Since A B does not depend on the value ω 0 ( j ), we bj ◦ B occurs if and only if ω( j ) = 1, and therefore the have in this case that A bj ◦ b conditional probability of A B is p j . When ω( j ) = 1, edge j is ‘contributing’ b b to either A j or B but not both. Replacing ω( j ) by ω 0 ( j ), we find similarly that bj +1 ◦ b the conditional probability of A B is at least p j . bj ◦ b In each of the three cases above, the conditional probability of A B is no b b greater than that of A j +1 ◦ B, and (4.17) follows.  c G. R. Grimmett 6 February 2009

[4.4]

Hoeffding inequality

49

4.4 Hoeffding inequality Let (Yn , Fn ), n ≥ 0, be a martingale. One can obtain bounds for the tail of Y n in terms of the sizes of the martingale differences Dk = Yk − Yk−1 . These bounds are surprisingly tight, and they have had substantial impact in various areas of application, especially those with a combinatorial structure. We describe such a bound in this section for the case when the Dk are bounded random variables. (4.18) Theorem. Hoeffding inequality. Let (Yn , Fn ), n ≥ 0, be a martingale such that |Yk − Yk−1 | ≤ K k (a.s.) for all k and some real sequence (K k ). Then
x > 0. P(Yn − Y0 ≥ x) ≤ exp − 21 x 2 /L n , P where L n = nk=1 K k2 .

Since Yn is a martingale, so is −Yn , and thus the same bound is valid for P(Yn − Y0 ≤ −x). Such inequalities are often named after Azuma [21] and Hoeffding [124]. See [161] for a review of the so-called ‘method of bounded differences’, and [109, Sect. 12.2], for some applications. Theorem 4.18 is one of a family of inequalities much used in probabilistic combinatorics, in what is termed the ‘method of bounded differences’. See the discussion in [161]. Its applications are of the following general form. Suppose that we are given N random variables X 1 , X 2 , . . . , X N , and we wish to study the behaviour of some function Z = Z (X 1 , X 2 , . . . , X N ). For example, the X i might be the sizes of objects to be packed in bins, and Z the minimum number of bins required to pack them. Let Fn = σ (X 1 , X 2 , . . . , X n ), and define the martingale Yn = E(Z | Fn ). Thus, Y0 = E(Z ) and Y N = Z . If the martingale differences are bounded, Theorem 4.18 provides a bound for the tail probability P(|Z − E(Z )| ≥ x). We shall see an application of this type at Theorem 11.13, which deals with the chromatic number of random graphs. Proof. The function g(d) = e ψd is convex for ψ > 0, and therefore (4.19)

eψd ≤ 12 (1 − d)e−ψ + 21 (1 + d)eψ

|d| ≤ 1.

Applying this to a random variable D having mean 0 and satisfying P(|D| ≤ 1) = 1, we obtain (4.20)

E(eψ D ) ≤ 12 (e−ψ + eψ ) < e 2 ψ , 1

2

ψ > 0.

where the final inequality follows by a comparison of the coefficients of the ψ 2n . By Markov’s inequality, (4.21)

P(Yn − Y0 ≥ x) ≤ e−θ x E(eθ (Yn −Y0 ) ),

With Dn = Yn − Yn−1 , E(eθ (Yn −Y0 ) ) = E(eθ (Yn−1 −Y0 ) eθ Dn ). c G. R. Grimmett 6 February 2009

θ > 0.

50

[4.5]

Correlation and Concentration

Since Yn−1 − Y0 is Fn−1 -measurable,

(4.22)

E(eθ (Yn −Y0 ) | Fn−1 ) = eθ (Yn−1 −Y0 ) )E(eθ Dn | Fn−1 ) ≤ eθ (Yn−1 −Y0 ) exp( 21 θ 2 K n2 ),

where we have applied (4.20) to the random variable Dn /K n at the last step. We take expectations of (4.22) and iterate to obtain E(eθ (Yn −Y0 ) ) ≤ E(eθ (Yn−1 −Y0 ) ) exp( 21 θ 2 K n2 ) ≤ exp( 12 θ 2 L n ).

Therefore, by (4.21),

P(Yn − Y0 ≥ x) ≤ exp(−θ x + 12 θ 2 L n ),

θ > 0.

Let x > 0, and set θ = x/L n (this is the value that minimizes the exponent). Then as required.

P(Yn − Y0 ≥ x) ≤ exp(− 12 x 2 /L n ),

x > 0,



4.5 Influence for product measures Let N ≥ 1 and E = {1, 2, . . . , N }, and write  = {0, 1} E . Let µ be a probability measure on , and A an event (that is, a subset of ). Two ways of defining the ‘influence’ of a member e ∈ E on the event A come to mind. The (conditional) influence is defined to be (4.23)

J A (e) = µ( A | ω(e) = 1) − µ( A | ω(e) = 0).

The absolute influence is (4.24)

I A (e) = µ(1 A (ωe ) 6= 1 A (ωe )),

where 1 A is the indicator function of A, and ω e , ωe are the configurations given by (4.3). In a voting analogy, each of N voters has 1 vote, and A is the set of vote-vectors that result in a given outcome. Then I A (e) is the probability that voter e can influence the outcome. We make two remarks concerning the above definitions. First, if A is increasing, (4.25) where

I A (e) = µ( Ae ) − µ( Ae ), Ae = {ω ∈  : ωe ∈ A},

Ae = {ω ∈  : ωe ∈ A}.

If, in addition, µ is a product measure, then I A (e) = J A (e). Note that influences depend on the underlying measure. Let φ p be product measure with density p on , and write φ = φ 1 , the uniform 2 measure. All logarithms are taken to base 2 until further notice. There has been extensive study of the largest (absolute) influence, max e I A (e), when µ is a product measure, and this has been used to obtain ‘sharp threshold’ theorems for the probability φ p ( A) of an increasing event A viewed as a function of p. c G. R. Grimmett 6 February 2009

[4.5]

Influence for product measures

51

(4.26) Theorem (Influence) [130]. There exists a constant c ∈ (0, ∞) such that the following holds. Let N ≥ 1, let E be a finite set with |E| = N , and let A be a subset of  = {0, 1} E with φ( A) ∈ (0, 1). Then X (4.27) I A (e) ≥ cφ( A)(1 − φ( A)) log[1/ max I A (e)], e

e∈E

where the reference measure is φ = φ 1 . There exists e ∈ E such that 2

(4.28)

I A (e) ≥ cφ( A)(1 − φ( A))

Note that φ( A)(1 − φ( A)) ≥

1 2

log N . N

min{φ( A), 1 − φ( A)}.

We indicate at this stage the reason why (4.27) implies (4.28). We may assume that m = maxe I A (e) satisfies m > 0, since otherwise φ( A)(1 − φ( A)) = 0. Since X I A (e) ≤ N m, e∈E

we have by (4.27) that

cφ( A)(1 − φ( A)) m ≥ . log(1/m) N Inequality (4.28) follows with an amended value of c, by the monotonicity of m/ log(1/m) or otherwise1 . Such results have applications to several topics including random graphs, random walks, and percolation, see [132]. We summarize two such applications next, and we defer until Section 5.8 a complete application to site percolation on the triangular lattice. I. First-passage percolation is concerned with passage times on a graph whose edges have random ‘travel-times’. Suppose we assign to each edge e of the ddimensional cubic lattice Ld a random travel-time Te , the Te being non-negative and independent with a common distribution function F. The passage time of a path π is the sum of the travel-times of its edges. Given two vertices u, v, the passage time Tu,v is defined as the infimum of the passage times of the set of paths joining u to v. The main question is to understand the asymptotic properties of T0,v as |v| → ∞. This model for the time-dependent flow of material was introduced in [117], and has been studied extensively since. It is a consequence of the subadditive ergodic theorem that, subject to a suitable moment condition, the (deterministic) limit 1 T0,nv n→∞ n

µv = lim 1 When

N = 1, there is nothing to prove. This is left as an exercise when N ≥ 2.

c G. R. Grimmett 6 February 2009

52

[4.5]

Correlation and Concentration

exists almost surely. Indeed, the subadditive ergodic theorem was conceived explicitly in order to prove such a statement for first-passage percolation. The constant µv is called the time constant in direction v. One of the open problems is to understand the asymptotic behaviour of var(T0,v ) as |v| → ∞. Various relevant results are known, and one of the best uses an influence theorem due to Talagrand [204], and related to Theorem 4.26. Specifically, it is proved in [28] that var(T0,v ) ≤ C|v|/ log |v| for some constant C = C(a, b, d), in the situation when each Te is equally likely to take either of the two positive values a, b. It has been predicted that var(T0,v ) ∼ |v|2/3 when d = 2. This work has been continued in [26]. II. The Voronoi percolation model is a continuum model that we construct as follows in R2 . Let 5 be a Poisson process of intensity 1 in R2 . With any u ∈ 5, we associate the ‘tile’ Tu = {x ∈ R2 : |x − u| ≤ |x − v| for all v ∈ 5}. Two points u, v ∈ 5 are declared adjacent, written u ∼ v, if Tu and Tv share a boundary segment. We now consider site percolation on the graph 5 with this adjacency relation. It was long believed that the critical percolation probability of this model is 21 (almost surely, with respect to the Poisson measure), and this was proved recently by Bollob´as and Riordan [42] using the threshold Theorem 4.78 that is consequent on Theorem 4.26. Bollob´as and Riordan showed also in [43] that a similar argument leads to an approach to the proof that the critical probability of bond percolation on Z 2 equals 1 2 . They used Theorem 4.78 in place of Kesten’s explicit proof of sharp threshold for this model, see [135, 136]. A “shorter” version of [43] is presented in Section 5.8 for the case of site percolation on the triangular lattice. We return to the influence theorem and its ramifications. There are several useful references concerning influence for product measures, see [84, 85, 130, 132] and their bibliographies2 . The order of magnitude N −1 log N is the best possible in (4.28), as shown by the following ‘tribes’ example taken from [27]. A population of N individuals comprises t ‘tribes’ each of cardinality s = log N − log log N + α. Each individual votes 1 with probability 21 and otherwise 0, and different individuals vote independently of one another. Let A be the event that there exists a tribe all of whose members vote 1. It is easily seen that 

1 1 − P( A) = 1 − s 2 s

t

α

∼ e−t/2 ∼ e−1/2 , 2 The treatment presented here makes heavy use of the work of the ‘Israeli’ school. The earlier paper of Russo [188] must not be overlooked, and there are several important papers of Talagrand [203, 204, 205, 206]. Later approaches to Theorem 4.26 can be found in [77, 183, 184].

c G. R. Grimmett 6 February 2009

[4.5]

Influence for product measures

53

and, for all i , 

1 I A (i ) = 1 − s 2 α

t−1

∼ e−1/2 2α−1

1 2s−1

log N , N

The ‘basic’ Theorem 4.26 on the discrete cube  = {0, 1} E can be extended to the ‘continuum’ cube K = [0, 1] E , and hence to other product spaces. We state the result for K next. Let λ be uniform (Lebesgue) measure on K . For a measurable subset A ⊆ K , it is usual (see, for example, [46]) to define the influence of e ∈ E on A as
L A (e) = λ N −1 {ω ∈ K : 1 A (ω) is a non-constant function of ω(e)} .

That is, L A (e) is the (N − 1)-dimensional Lebesgue measure of the set of all ψ ∈ [0, 1] E \{e} with the property that: both A and its complement A intersect the ‘fibre’ Fψ = {ψ} × [0, 1] = {ω ∈ K : ω( f ) = ψ( f ), f 6= e}.

It is more natural to consider elements ψ for which A ∩ Fψ has Lebesgue measure strictly between 0 and 1, and thus we define the influence in these notes by (4.29)


I A (e) = λ N −1 {ψ ∈ [0, 1] E \{e} : 0 < λ1 ( A ∩ Fψ ) < 1} .

Here and later, we write λk for k-dimensional Lebesgue measure. Note that I A (e) ≤ L A (e). (4.30) Theorem [46]. There exists a constant c ∈ (0, ∞) such that the following holds. Let N ≥ 1, let E be a finite set with |E| = N , and let A be an increasing subset of the cube K = [0, 1] E with λ( A) ∈ (0, 1). Then (4.31)

X e∈E

I A (e) ≥ cλ( A)(1 − λ( A)) log[1/(2m)],

where m = maxe I A (e), and the reference measure on K is Lebesgue measure λ. There exists e ∈ E such that (4.32)

I A (e) ≥ cλ( A)(1 − λ( A))

log N . N

We shall see in Theorem 4.35 that the condition of monotonicity of A can be removed. The factor ‘2’ in (4.31) is innocent in the following regard. The inequality is important only when m is small, and, for m ≤ 31 say, one may remove the ‘2’ and replace c by a larger constant. c G. R. Grimmett 6 February 2009

54

Correlation and Concentration

[4.5]

Results similar to those of Theorems 4.26 and 4.30 have been proved in [89] for certain non-product measures, and all increasing events. Let µ be a positive probability measure on the discrete space  = {0, 1} E satisfying the FKG lattice condition (4.12). For any increasing subset A of  with µ( A) ∈ (0, 1), we have that X (4.33) J A (e) ≥ cµ( A)(1 − µ( A)) log[1/(2m)], e∈E

where m = maxe J A (e). Furthermore, as above, there exists e ∈ E such that log N . (4.34) J A (e) ≥ cµ( A)(1 − µ( A)) N Note the use of conditional influence J A (e), with reference measure µ. Indeed, (4.34) can fail for all e when J A is replaced by I A . The proof of (4.33) makes use of Theorem 4.30, and is omitted here, see [89, 90]. The domain of Theorem 4.30 can be extended to powers of an arbitrary probability space, that is with ([0, 1], λ1 ) replaced by a general probability space. Let |E| = N and let X = (6, F , P) be a probability space. We write X E for the product space of X . Let A ⊆ 6 E be measurable. The influence of e ∈ E is given as in (4.29) by
I A (e) = P {ψ ∈ 6 E \{e} : 0 < P( A ∩ Fψ ) < 1} ,

with P = P E and Fψ = {ψ}×6, the ‘fibre’ of all ω ∈ X E such that ω( f ) = ψ( f ) for f 6= e. The following theorem contains two statements: that the influence inequalities are valid for general product spaces, and that they hold for non-increasing events. We shall require a condition on X = (6, F , P) for the first of these, and we state this next 3 . The pair (F , P) generates a measure ring (see [113, §40] for the relevant definitions). We call this measure ring separable if it is separable when viewed as a metric space with metric ρ(B, B 0 ) = P(B 4 B 0 ). (4.35) Theorem [46]. Let X = (6, F , P) be a probability space whose nonatomic part is separable. Let N ≥ 1, let E be a finite set with |E| = N , and let A ⊆ 6 E be measurable in the product space X E , with P( A) ∈ (0, 1) . There exists an absolute constant c ∈ (0, ∞) such that: X (4.36) I A (e) ≥ c P( A)(1 − P( A)) log[1/(2m)], e∈E

where m = maxe I A (e), and the reference measure is P = P E . There exists e ∈ E with log N . (4.37) I A (e) ≥ c P( A)(1 − P( A)) N Of especial interest is the case when 6 = {0, 1} and P is Bernoulli measure with density p. Note that the atomic part of X is always separable, since there can be at most countably many atoms. 3 This

condition is omitted from [46].

c G. R. Grimmett 6 February 2009

[4.6]

Proofs of influence theorems

55

4.6 Proofs of influence theorems This section contains the proofs of the theorems of the last. Proof of Theorem 4.26. We use a Fourier analysis of functions f :  → R. Define the inner product by f, g :  → R,

h f, gi = φ( f g),

where φ = φ 1 , so that the L 2 -norm of f is given by 2

q p k f k2 = φ( f 2 ) = h f, f i.

We call f Boolean if it takes values in the set {0, 1}. Boolean functions are in one–one correspondence with the power set of E via the relation f = 1 A ↔ A. If f is Boolean, say f = 1 A , then k f k22 = φ( f 2 ) = φ( f ) = φ( A).

(4.38) For F ⊆ E, let u F (ω) =

Y

e∈F

(−1)ω(e) = (−1)

e∈F

ω(e)

,

ω ∈ .

It can be checked that the functions u F , F ⊆ E, form an orthonormal basis for the function space. Thus, for f :  → R, f =

X

F ⊆E

fˆ(F)u F ,

where the so-called Fourier–Walsh coefficients of f are given by fˆ(F) = h f, u F i,

F ⊆ E.

In particular, fˆ(∅) = φ( f ),

and

h f, gi =

X

F ⊆E

fˆ(F)g(F), ˆ

and the latter yields the Parseval relation (4.39)

k f k22 =

c G. R. Grimmett 6 February 2009

X

F ⊆E

fˆ(F)2 .

56

[4.6]

Correlation and Concentration

Fourier analysis operates harmoniously with influences as follows. For f = 1 A and e ∈ E, let f e (ω) = f (ω) − f (κe ω),

where κe ω is the configuration ω with the state of e flipped. Since f e takes values in the set {−1, 0, +1}, we have that | f e | = f e2 . The Fourier–Walsh coefficients of f e are given by X 1   f (ω) − f (κe ω) (−1)|B∩F | N 2 ω∈ X 1   = f (ω) (−1)|B∩F | − (−1)|(B 4 {e})∩F | , N 2

fˆe (F) = h f e , u F i =

ω∈

where B = η(ω) := {e ∈ E : ω(e) = 1} is the set of ω-open indices. Now,    0 if e ∈ / F, |B∩F | |(B 4 {e})∩F | (−1) − (−1) = |B∩F | 2(−1) = 2u F (ω) if e ∈ F, so that (4.40)

fˆe (F) =



0 2 fˆ(F)

if e ∈ / F,

if e ∈ F.

The influence I (e) = I A (e) is the mean of | f e | = f e2 , whence, by (4.39), (4.41)

I (e) = k f e k22 = 4

X

F : e∈F

fˆ(F)2 ,

and the total influence is (4.42)

X e∈E

I (e) = 4

X

F ⊆E

|F| fˆ(F)2 .

P 2 ˆ We propose to find an upper bound for the sum φ( A) = F f (F) . From (4.42) we will extract an upper bound for the contributions to this sum from the fˆ(F)2 for large |F|. This will be combined with a corresponding estimate for small |F| that will be obtained as follows by considering a re-weighted sum P 2 2|F | for 0 < ρ < 1. ˆ F f (F) ρ For w ∈ [1, ∞), we define the L w -norm kgkw = φ(|g|w )1/w ,

g :  → R,

recalling that kgkw is non-decreasing in w. For ρ ∈ R, let Tρ g be the function Tρ g = c G. R. Grimmett 6 February 2009

X

F ⊆E

|F | g(F)ρ ˆ uF

[4.6]

Proofs of influence theorems

so that

kTρ gk22 =

X

F ⊆E

57

2 2|F | g(F) ˆ ρ .

When ρ ∈ [−1, 1], Tρ g has a probabilistic interpretation. For ω ∈ , let 9 = (9(e) : e ∈ E) be a random vector such that: the 9(e), e ∈ E, are independent, and 9(e) =



ω(e) 1 − ω(e)

with probability 21 (1 + ρ),

otherwise.

We claim that (4.43)

Tρ g(ω) = E(g(9)),

thus explaining why Tρ is sometimes called the ‘noise operator’. Equation (4.43) is proved as follows. First, for F ⊆ E, E(u F (9)) = E

=

Y

Now, g =

P

F

e∈F

Y

(−1)ω(e)

e∈F |F |

=ρ

(−1)

9(e)

u F (ω).

1



1 2 (1 + ρ) − 2 (1 − ρ)



g(F)u ˆ F , so that E(g(9)) =

=

X

F ⊆E

X

F ⊆E

g(F) ˆ E(u F (9)) |F | g(F)ρ ˆ u F (ω) = Tρ g(ω),

as claimed at (4.43). The next proposition is pivotal for the proof of the theorem. It is sometimes referred to as the ‘hypercontractivity’ lemma, and it is related to the log-Sobolev inequality. It is commonly attributed to subsets of Bonami [45], Gross [112], Beckner [24], each of whom has worked on estimates of this type. The proof is omitted. (4.44) Proposition. For g :  → R and ρ > 0, kTρ gk2 ≤ kgk1+ρ 2 . c G. R. Grimmett 6 February 2009

58

[4.6]

Correlation and Concentration

Let 0 < ρ < 1. Set g = f e where f = 1 A , noting that g takes the values 0, ±1 only. Then, X

F : e∈F

4 fˆ(F)2 ρ 2|F | =

X

F ⊆E

fˆe (F)2 ρ 2|F |

by (4.40)

= kTρ f e k22

 2 2/(1+ρ 2 ) ≤ k f e k21+ρ 2 = φ(| fe |1+ρ ) 4/(1+ρ 2 )

= k f e k2

= I (e)2/(1+ρ

2

)

by Proposition 4.44 by (4.41).

Therefore, X

(4.45)

I (e)2/(1+ρ

e∈E

2

)

≥4

X

F ⊆E

Let t = φ( A) = fˆ(∅). By (4.45), X 2 (4.46) I (e)2/(1+ρ ) ≥ 4ρ 2b e∈E

= 4ρ

2b

|F| fˆ(F)2 ρ 2|F | .

X

0<|F |≤b

X

|F |≤b

fˆ(F)2 2

fˆ(F) − t

2



,

where b ∈ (0, ∞) will be chosen later. By (4.42), X X I (e) ≥ 4b fˆ(F)2 , |F |>b

e∈E

which we add to (4.46) to obtain (4.47)

ρ −2b

X

I (e)2/(1+ρ

2

)

e∈E

+

X 1X fˆ(F)2 − 4t 2 I (e) ≥ 4 b F ⊆E

e∈E

= 4t (1 − t)

by (4.39).

We are now ready to prove (4.27). Let m = max e I (e), noting that m > 0 since φ( A) 6= 0, 1. The claim is trivial if m = 1, and we assume that m < 1. Then X X I (e)4/3 ≤ m 1/3 I (e), e∈E

e∈E

whence, by (4.47) and the choice ρ 2 = 21 , (4.48)



2 m b

1/3

c G. R. Grimmett 6 February 2009

 1 X I (e) ≥ 4t (1 − t). + b e∈E

[4.6]

Proofs of influence theorems

59

We choose b such that 2b m 1/3 = b−1 , and it is an easy exercise that b ≥ A log(1/m) for some absolute constant A > 0. With this choice of b, (4.27) follows from (4.48) with c = 2 A. Inequality (4.32) follows, as explained after the statement of the theorem.  Proof of Theorem 4.30. We follow [84]. The idea of the proof is to ‘discretize’ the cube K and the increasing event A, and to apply Theorem 4.26. Let k ∈ {1, 2, . . . } to be chosen later, and subdivide the N -cube K = [0, 1] E into 2k N disjoint smaller cubes each of side-length 2−k . These small cubes are of the form Y (4.49) B(l) = [le , le + 2−k ), e∈E

where l = (le : e ∈ E) and each le is a ‘binary decimal’ of the form l e = 0.le,1le,2 · · · le,k with each le, j ∈ {0, 1}. There is a special case. When l e = 0.11 · · · 1, we put the closed interval [l e , le + 2−k ] into the product of (4.49). Lebesgue measure λ on K induces product measure φ with density 12 on the space  = {0, 1}k N of 0/1-vectors (le, j : j = 1, 2, . . . , k, e ∈ E). We call each B(l) a ‘small cube’. We claim that it suffices to consider events A that are the unions of small cubes. For a measurable subset A ⊆ K , let Aˆ be the subset of K that ‘approximates’ to S A, given by Aˆ = l∈A B(l) where A = {l ∈  : B(l) ∩ A 6= ∅}.

Note that A is an increasing subset of the discrete k N -cube . We write I A (e, j ) for the influence of the index (e, j ) on the subset A ⊆  under the measure φ. ˆ the measure and influences The next task is to show that, when replacing A by A, of A are not greatly changed. (4.50) Lemma [46]. In the above notation, (4.51) (4.52)

ˆ − λ( A) ≤ N , 0 ≤ λ( A) 2k 2N |I Aˆ (e) − I A (e)| ≤ k , 2

e ∈ E.

ˆ whence λ( A) ≤ λ( A). ˆ Let µ : K → K be the projection Proof. Clearly A ⊆ A, mapping that maps (x f : f ∈ E) to (x f − m : f ∈ E) where m = ming∈E x g . We have that (4.53)

ˆ − λ( A) ≤ |R|2−k N , λ( A)

where R is the set of small cubes that intersect both A and its complement A. Since A is increasing, R cannot contain two distinct elements r , r 0 with µ(r ) = µ(r 0 ). c G. R. Grimmett 6 February 2009

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Correlation and Concentration

Therefore, |R| is no larger than the number of faces of small cubes lying in the ‘hyperfaces’ of K , that is, (4.54)

K ≤ N 2k(N −1) .

Inequality (4.51) follows by (4.53). Let e ∈ E, and let µe : K → [0, 1] E \{e} be the projection that sends (x f : f ∈ E) to (x f : f ∈ E \ {e}). The face µe (K ) is the union of ‘small faces’ of small cubes. Each small face F corresponds to a ‘tube’ T (F) of small cubes, based on that face with axis parallel to the eth direction. See Figure 4.1. Such a tube has ‘first’ face F and ‘last’ face L = T (F) ∩ {ω ∈ K : ω(e) = 1}, and we write B F (respectively, B L ) for the (unique) small cube with face F (respectively, L). It is easily seen that F contributes 0 to I Aˆ (e) − I A (e) if 1 A is constant on both B F and B L (it is not important that 1 A should take the same value on the initial small cube as on the final). Therefore, (4.55)

|I Aˆ (e) − I A (e)| ≤ |N F ∪ N L |2−k(N −1) ,

where N F (respectively, N L ) is the set of initial (respectively, final) small cubes S on which 1 A is non-constant. By restricting 1 A to the ‘fattened hyperface’ {B F : F ⊆ µe (K )} and applying the argument leading to (4.54) within this region, we find as there that |N F | ≤ (N − 1)2k(N −2) .

The same inequality holds with N L in place of N F , and inequality (4.52) follows by (4.55). 

Let A be an increasing subset of K , assume 0 < t = λ( A) < 1, and let m = maxe I A (e). We may assume that 0 < m < 21 , since otherwise (4.31) is a triviality. With Aˆ given as above for some value of k to be chosen soon, we write ˆ and mˆ = maxe I ˆ (e). We shall prove below that tˆ = λ( A) A X

(4.56)

e∈E

I Aˆ (e) ≥ ctˆ(1 − tˆ) log[1/(2m)], ˆ

for some absolute constant c > 0. Let k = k(N , A) be sufficiently large that the following inequalities hold:   1 m log[1/(2m)] 1 N (4.57) ≤ min t (1 − t), , −m , 2k 2 2 + log[1/(2m)] 2 (4.58)

2N 2 1 ≤ ct (1 − t) log[1/(2m)]. 2k 8

By Lemma 4.50, (4.59)

|t − tˆ| ≤

c G. R. Grimmett 6 February 2009

N , 2k

|m − m| ˆ ≤

2N , 2k

[4.6]

Proofs of influence theorems

r

BF

B

61

BL L

s

ω(e)

Figure 4.1. The small boxes B = B(r, s) form the tube T (r). The region A is shaded.

whence, by (4.57)–(4.58), (4.60) |t − tˆ| ≤ 12 t (1 − t),

|m − m| ˆ ≤ m ∧ mˆ

mˆ < 21 ,

By Lemma 4.50 again, X e∈A

I A (e) ≥

X e∈A

I Aˆ (e) −

1 2

log[1/(2m)].

2N 2 . 2k

By (4.56), (4.58), and (4.60), X e∈A



I A (e) ≥ c t (1 − t) − |t − tˆ|





 2N 2 |m − m| ˆ − k log[1/(2m)] − m ∧ mˆ 2

≥ 18 ct (1 − t) log[1/(2m)]

as required. It thus suffices to prove (4.56), and we shall henceforth assume that A is a union of small cubes. (4.61) Lemma [46, 84]. For e ∈ E, k X j =1

c G. R. Grimmett 6 February 2009

IA (e, j ) ≤ 2I A (e).

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Correlation and Concentration

Proof. Let e ∈ E. For a fixed vector r = (r 1 , r2 , . . . , r N −1 ) ∈ ({0, 1}k ) E \{e} , consider the ‘tube’ T (r) comprising the union of the small cubes B(r, s) over the 2k possible values in s ∈ {0, 1}k . One sees after a little thought (see Figure 4.1) that X IA (e, j ) = ( 21 )k N −1 K (r, j ), r

where K (r, j ) is the number of unordered pairs S = B(r, s), S 0 = B(r, s 0 ) of small cubes of T (r) such that: S ⊆ A, S 0 6⊆ A, and |s − s 0 | = 2− j . Since A is an increasing subset of K , one can see that K (r, j ) ≤ 2k− j , whence

X j

IA (e, j ) ≤

2k 2k N −1

j = 1, 2, . . . , k, JN =

2 2k(N −1)

JN ,

where J N is the number of tubes T (r) that intersect both A and its complement A. Now, 1 I A (e) = k(N −1) J N , 2 and the lemma is proved.  We return to the proof of (4.56). The c j that follow are absolute positive constants. Assume that m = maxe I A (e) < 21 . By Lemma 4.61, IA (e, j ) ≤ 2m

for all e, j.

By (4.27) applied to the event A of the k N -cube , X IA (e, j ) ≥ c2 t (1 − t) log[1/(2m)], e, j

where t = λ( A). By Lemma 4.61 again, X I A (e) ≥ 21 c2 t (1 − t) log[1/(2m)], e∈E

as required at (4.56).



Proof of Theorem 4.35. We prove this in two steps. I. In the notation of the theorem, there exists a Lebesgue-measurable subset B of K = [0, 1] E such that: P( A) = λ(B), and I A (e) ≥ I B (e) for all e, where the influences are calculated according to the appropriate probability measures. II. There exists an increasing subset C of K such that λ(B) = λ(C), and I B (e) ≥ IC (e) for all e. c G. R. Grimmett 6 February 2009

[4.6]

Proofs of influence theorems

63

The claims of the theorem follow via Theorem 4.30 from these two facts. A version of Claim I was stated in [46] without proof. We use the measurespace isomorphism theorem 4 , Theorem B of [113, p. 173] (see also [1, p. 3] or [174, p. 16]). Let x 1 , x 2 , . . . be an ordering of the atoms of X , and let Ii be the sub-interval [qi , qi +1 ) of [0, 1], where qi =

i −1 X

P({x j }).

j =1

The non-atomic part of X has sample space 6 0 = 6 \ {x 1 , x 2 , . . . }, and total measure 1 − q∞ . By the isomorphism theorem, there exists a measure-preserving map µ from the σ -algebra F 0 of 6 0 to the Borel σ -algebra of the interval [q∞ , 1] endowed with Lebesgue measure λ1 , satisfying λ

µ( A1 \ A2 ) = µ A1 \ µ A2 , [  [ ∞ ∞ λ µ An = µ An ,

(4.62)

n=1

n=1

for An ∈ F 0 , where A = B means that λ1 ( A 4 B) = 0. We extend the domain of µ to F by setting µ({x i }) = Ii . In summary, there exists µ : F → B[0, 1] such that P( A) = λ1 (µ A) for A ∈ F , and (4.62) holds for A n ∈ F . The product σ Q -algebra F E of X E is generated by the class R E of ‘rectangles’ of the form R = e∈E Ae for Ae ∈ F . For such R ∈ R E , let Y µE R = µ Ae . λ

e∈E

We extend the domain of µ E to the class U of finite unions of rectangles by µ

E

[ m

i =1

Ri



=

m [

µ E Ri .

i =1

It can be checked that (4.63)

P(R) = λ E (µ E R),

for any such union R. Let A ∈ F E . We can find an increasing sequence (Un : n ≥ 1) of elements of U, each being a union of rectangles with strictly positive measure, such that P( A 4 Un ) → 0 as n → ∞, and hence (4.64) 4 Tom

P(Un \ A) = 0. Liggett kindly proposed the use of the isomorphism theorem.

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Correlation and Concentration

Let Vn = µ E Un and B = limn→∞ Vn . Since Vn is non-decreasing in n, by (4.63), λ E (B) = lim λ E (µ E Un ) = lim P(Un ) = P( A). n→∞

n→∞

We turn now to the influences. Let e ∈ E, and


J Aa = P E \{e} {ψ ∈ 6 E \{e} : P( A ∩ Fψ ) = a} ,

a = 0, 1,

where Fψ = {ψ} × 6 is the ‘fibre’ at ψ. We define J Ba similarly, with P replaced by λ and Fψ replaced by the fibre {ψ} × [0, 1] of K . Thus, (4.65)

I A (e) = 1 − J A0 − J A1 ,

and we claim that J A0 ≤ J B0 .

(4.66)

By replacing A by its complement A, we obtain that J A1 ≤ J B1 , and it follows by (4.65)–(4.66) that I A (e) ≥ I B (e), as required. We write Un as the finite union S Un = i Fi × G i where each Fi (respectively, G i ) is a rectangle of 6 E \{e} (respectively, 6). By Fubini’s theorem and (4.64), [  0 0 E \{e} J A ≤ JUn = 1 − P Fi i

=1−λ

E \{e}

[

µ

E \{e}

i

Fi



= JV0n ,

by (4.63) with E replaced by E \ {e}. Finally, we show that J V0n → J B0 as n → ∞, and (4.66) will follow. For ψ ∈ 6 E , we write proj(ψ) for the projection of ψ onto the sub-space 6 E \{e} . Since the Vn are unions of rectangles of [0, 1] E with strictly positive measure,
JV0n = λ E \{e} proj Vn .

Now, Vn ↑ B, so that proj Vn ↓ proj B, whence JV0n → λ E \{e} (proj B). Using the fact that ψ ∈ B if and only if ψ ∈ Vn for some n, we have that λ E \{e} (proj B) = J B0 , and (4.66) follows. Claim I is proved. Claim II is proved by an elaboration of the method laid out in [27, 46]. Let B ⊆ K be a non-increasing event. For e ∈ E and ψ = (ω(g) : g 6= e) ∈ [0, 1] E \{e}, we define the fibre Fψ as usual by Fψ = {ψ} × [0, 1]. We replace B ∩ Fψ by the set  {ψ} × (1 − y, 1] if y > 0, (4.67) Bψ = ∅ if y = 0, c G. R. Grimmett 6 February 2009

[4.6]

Proofs of influence theorems

65

f

ψ

B ∩ Fψ

Bψ

e

ω

ω

Figure 4.2. In the e/ f -plane, we push every B ∩ Fψ as far rightwards along the fibre Fψ as possible.

where (4.68)

y = y(ψ) = λ1 (B ∩ Fψ ).

Thus Bψ is obtained from B by ‘pushing B ∩ Fψ up the fibre’ in a measureS preserving manner. See Figure 4.2. Clearly, Me B = ψ Bψ is increasing5 in the direction e and, by Fubini’s theorem, (4.69)

λ(Me B) = λ(B).

We order E in an arbitrary manner, and let Y  C= Me B, e∈E

where the product is constructed in the given order. By (4.69), λ(C) = λ(B). We show that C is increasing by proving that: if B is increasing in direction f ∈ E where f 6= e, then so is Me B. It is enough to work with the reduced sample space K 0 = [0, 1]{e, f } , as illustrated in Figure 4.2. Suppose that ω, ω 0 ∈ K 0 are such that ω(e) = ω0 (e) and ω( f ) < ω0 ( f ). Then  1 if ω(e) > 1 − y, (4.70) 1 Me B (ω) = 0 if ω(e) ≤ 1 − y,

where y = y(ω( f )) is given according to (4.68), with a similar expression with ω and y replaced by ω 0 and y 0 . Since B is assumed increasing in ω( f ), we have that y ≤ y 0 . By (4.70), if ω ∈ Me B, then ω0 ∈ Me B, which is to say that Me B is increasing in direction f . Finally, we show that (4.71) 5 Exercise:

I Me B ( f ) ≤ I B ( f ), Show that Me B is Lebesgue-measurable.

c G. R. Grimmett 6 February 2009

f ∈ E,

66

[4.7]

Correlation and Concentration

whence IC ( f ) ≤ I B ( f ) and the theorem is proved. First, by construction, I Me B (e) = I B (e). Let f 6= e. By conditioning on ω(g) for g 6= e, f , 
 I Me B ( f ) = λ E \{e, f } λ1 {ω(e) : 0 < λ1 (Me B ∩ Fν ) < 1}

where ν = (ω(g) : g 6= f ) and Fν = {ν} × [0, 1]. We shall show that (4.72)

λ1 {ω(e) : 0 < λ1 (Me B ∩ Fν ) < 1} ≤ λ1 {ω(e) : 0 < λ1 (B ∩ Fν ) < 1} ,

and the claim will follow. Inequality (4.72) depends only on ω(e), ω( f ), and thus we shall make no further reference to the remaining coordinates ω(g), g 6= e, f . Henceforth, we write ω for ω(e) and ψ for ω( f ). With the aid of Figure 4.2, we see that the left side of (4.72) equals ω − ω, where ω = sup{ω : λ1 (Me B ∩ Fω ) < 1},

(4.73)

ω = inf{ω : λ1 (Me B ∩ Fω ) > 0}.

Let  be positive and small, and let (4.74)

A = {ψ : λ1 (B ∩ Fψ ) > 1 − ω − }.

Since λ1 (B ∩ Fψ ) = λ1 (Me B ∩ Fψ ), λ1 ( A ) > 0 by (4.73). Let A 0 = [0, 1]× A . We estimate the two-dimensional Lebesgue measure λ2 (B ∩ A0 ) in two ways: λ2 (B ∩ A0 ) > λ1 ( A )(1 − ω − )

by (4.74),

λ2 (B ∩ A0 ) ≤ λ1 ( A )λ1 ({ω : λ1 (B ∩ Fω ) > 0}), whence C = {ω : λ1 (B ∩ Fω ) > 0} satisfies λ1 (C) ≥ lim[1 − ω − ] = 1 − ω. ↓0

By a similar argument, D = {ω : λ1 (B ∩ Fω ) = 1} satisfies λ1 (D) ≤ 1 − ω. For ω ∈ C \ D, 0 < λ1 (B ∩ Fω ) < 1, so that I B (e) ≥ λ1 (C \ D) ≥ ω − ω, and (4.71) follows.

c G. R. Grimmett 6 February 2009



[4.7]

Russo formula, and sharp thresholds

67

4.7 Russo formula, and sharp thresholds Let φ p denote product measure with density p on the finite product space  = {0, 1} E . The influence I A (e), of e ∈ E on an event A, is given in (4.25). (4.75) Theorem. Russo formula. For any event A ⊆ , X X d I A (e). [φ p ( Ae ) − φ( Ae )] = φ p ( A) = dp e∈E

e∈E

This formula, or its equivalent, has been discovered by a number of authors. See, for example, [23, 160, 187]. The element e ∈ E is called pivotal for the event A if the occurrence or not of A depends on the state of e, that is, if 1 A (ωe ) 6= 1 A (ωe ). If A is increasing, Russo’s formula states that φ 0p ( A) equals the mean number of pivotal elements of E. Proof. This is standard, see for example [95]. It is elementary that (4.76)

 X  |η(ω)| d N − |η(ω)| 1 A (ω)φ p (ω), φ p ( A) = − dp p 1− p ω∈

where η(ω) = {e ∈ E : ω(e) = 1} and N = |E|. Setting A = , we find that 1 φ p (|η| − p N ), p(1 − p)

0= whence p(1 − p)


d φ p ( A) = φ p [|η| − p N ]1 A − φ p (|η| − p N )φ p (1 A ) dp = φ p (|η|1 A ) − φ p (|η|)φ p (1 A ) X  = φ p (1e 1 A ) − φ p (1e )φ p (1 A ) , e∈E

where 1e is the indicator function that e is open. The summand equals   pφ p ( Ae ) − p pφ p ( Ae ) + (1 − p)φ p ( Ae ) .

and the formula is proved.



Let A be an increasing event in  = {0, 1} E that is non-trivial in that A 6= ∅, . The function f ( p) = φ p ( A) is non-decreasing with f (0) = 0 and f (1) = 1. The next theorem is an immediate consequence of Theorems 4.35 and 4.75. c G. R. Grimmett 6 February 2009

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[4.7]

Correlation and Concentration

(4.77) Theorem [204]. There exists a constant c > 0 such that the following holds. Let A be an increasing subset of  with A 6= ∅, . Then, for p ∈ (0, 1), d φ p ( A) ≥ cφ p ( A)(1 − φ p ( A)) log[1/(2 max I A (e)], e dp where I A (e) is the influence of e on A with respect to the measure φ p . Theorem 4.77 takes an especially simple form when A has a certain property of symmetry. In such a case, the following sharp-threshold theorem implies that f ( p) = φ p ( A) increases from (near) 0 to (near) 1 over an interval of p-values with length of order not exceeding 1/ log N . Let 5 be the group of permutations of E. Any π ∈ 5 acts on  by π ω = (ω(πe ) : e ∈ E). We say that a subgroup A of 5 acts transitively on E if, for all pairs j, k ∈ E, there exists α ∈ A with α j = k. Let A be a subgroup of 5. A probability measure φ on (, F ) is called Ainvariant if φ(ω) = φ(αω) for all α ∈ A. An event A ∈ F is called A-invariant if A = α A for all α ∈ A. It is easily seen that, for any subgroup A, φ p is A-invariant. (4.78) Theorem (Sharp threshold) [85]. There exists a constant c satisfying c ∈ (0, ∞) such that the following holds. Let N = |E| ≥ 1. Let A ∈ F be an increasing event, and suppose there exists a subgroup A of 5 acting transitively on E such that A is A-invariant. Then (4.79)

d φ p ( A) ≥ cφ p ( A)(1 − φ p ( A)) log N , dp

p ∈ (0, 1).

Let  ∈ (0, 21 ) and let A be increasing and non-trivial (in the above sense). Under the conditions of the theorem, φ p ( A) increases from  to 1 −  over an interval of values of p having length of order not exceeding 1/ log N . This amounts to a quantification of the so-called S-shape results described and cited in [95, Sect. 2.5]. An early step in the direction of sharp thresholds was taken by Russo [188] (see also [204]), but without the quantification of log N . Essentially the same conclusions hold for a family {µ p : p ∈ (0, 1)} of probability measures given as follows in terms of a positive measure µ satisfying the FKG lattice condition. For p ∈ (0, 1), let µ p be given by   1 Y ω(e) 1−ω(e) (4.80) µ p (ω) = p (1 − p) µ(ω), ω ∈ , Zp e∈E

where Z p is chosen in such a way that µ p is a probability measure. It is easy to check that each µ p satisfies the FKG lattice condition. It turns out that, for an increasing event A 6= ∅, , (4.81)

cξ p d µ p ( A) ≥ µ p ( A)(1 − µ p ( A)) log[1/(2 max J A (e)], e dp p(1 − p)

c G. R. Grimmett 6 February 2009

[4.8]

where

Exercises

69

  ξ p = min µ p (ω(e) = 1)µ p (ω(e) = 0) . e∈E

The proof uses inequality (4.33) and proceeds as in [89]. This extension of Theorem 4.77 does not appear to have been noted before. It may be used in the studies of the random-cluster model, and of the Ising model with external field (see [90]). A slight variant of Theorem 4.78 is valid for measures φ p given by (4.80), with the positive probability measure µ satisfying: µ satisfies the FKG lattice condition, and µ is A-invariant. See (4.81) and [89, 98]. From amongst the issues arising from the sharp-threshold Theorem 4.78, we identify two. First, to what degree is information about the group A relevant to the sharpness of the threshold. Secondly, what can be said when p = p N tends to 0 as N → ∞. The reader is referred to [132] for answers to these questions.

Proof of Theorem 4.78. We show first that the influences I A (e) are constant for e ∈ E. Let e, f ∈ E, and find α ∈ A such that αe = f . Under the given conditions, X X φ p ( A, 1 f = 1) = φ p (ω)1 f (ω) = φ p (αω)1e (αω) ω∈A

=

X

ω0 ∈A

ω∈A

φ p (ω0 )1e (ω0 ) = φ p ( A, 1e = 1),

where 1g is the indicator function that ω(g) = 1. On setting A = , we deduce that φ p (1 f = 1) = φ p (1e = 1). On dividing, we obtain that φ p ( A | 1 f = 1) = φ p ( A | 1e = 1). A similar equality holds with 1 replaced by 0, and therefore I A (e) = I A ( f ). It follows that X I A ( f ) = N I A (e). f ∈E

By Theorem 4.35 applied to the product space (, F , φ p ), the right side is at least cφ p ( A)(1 − φ p ( A)) log N , and (4.79) is a consequence of Theorem 4.75. 

4.8 Exercises 4.1. Let X n , Yn ∈ L 2 (, F , P) be such that X n → X , Yn → Y in L 2 . Show that X n Yn → X Y in L 1 . [Reminder: L p is the set of random variables Z with E(|Z | p ) < ∞, and Z n → Z in L p if E(|Z n − Z | p ) → 0. You may use any standard fact such as the Cauchy–Schwarz inequality.] 4.2. [121] Let P p be a product measure on the space {0, 1}n with density p. Show by induction on n that P p satisfies the Harris–FKG inequality, which is to say that P p ( A ∩ B) ≥ P p ( A)P p (B) for any pair A, B of increasing events. c G. R. Grimmett 6 February 2009

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[4.8]

Correlation and Concentration

4.3. (continuation) Consider bond percolation on the square lattice Z 2 . Let X and Y be increasing functions on the sample space, such that P p (X 2 ), P p (Y 2 ) < ∞. Show that X and Y are positively associated. 4.4. Coupling. (a) Take  = [0, 1], with the Borel σ -field and Lebesgue measure P. For any distribution function F, define a random variable Z F on  by Z F (ω) = inf {z : ω ≤ F(z)}, Prove that

ω ∈ .




P(Z F ≤ z) = P [0, F(z)] = F(z),

whence Z F has distribution function F. (b) For real-valued random variables X , Y , we write X ≤st Y if P(X ≤ u) ≥ P(Y ≤ u) for all u. Show that X ≤st Y if and only if there exist random variables X 0 , Y 0 on , with the same respective distributions as X and Y , such that P(X 0 ≤ Y 0 ) = 1. 4.5. [98] Let µ be a positive probability measure on the finite product  = {0, 1} E . Show that it satisfies the FKG lattice condition µ(ω1 ∨ ω2 )µ(ω1 ∧ ω2 ) ≥ µ(ω1 )µ(ω2 ),

ω1 , ω2 ∈ ,

if and only if this inequality holds for all pairs ω1 , ω2 that differ on exactly two elements of E. 4.6. [98] Let µ1 , µ2 be positive probability measures on the finite product  = {0, 1} E . Assume that they satisfy µ2 (ω1 ∨ ω2 )µ1 (ω1 ∧ ω2 ) ≥ µ1 (ω1 )µ2 (ω2 ), for all pairs ω1 , ω2 ∈  that differ on either one or two elements of E, and in addition that µ1 satisfies the FKG lattice condition. Show that µ2 ≥st µ1 . 4.7. Let X 1 , X 2 , . . . be independent Bernoulli random variables with parameter p, and Sn = X 1 + X 2 + · · ·+ X n . Show by Hoeffding’s inequality or otherwise that √
x > 0, P |Sn − np| ≥ x n ≤ 2 exp(− 21 x 2 /m),

where m = min{ p, 1 − p}. 4.8. Let G n, p be the random graph with vertex set V = {1, 2, . . . , n} obtained by joining each pair of distinct vertices by an edge with probability p (different pairs are joined independently). Show that the chromatic number χ n, p satisfies
P |χn, p − Eχn, p | ≥ x ≤ 2 exp(− 12 x 2 /n), x > 0. 4.9. Russo’s formula. Let X be a random variable on the finite sample space  = {0, 1} E . Show that X d P p (X ) = P p (δe X ) dp e∈E

c G. R. Grimmett 6 February 2009

[4.8]

Exercises

71

where δe X (ω) = X (ωe ) − X (ωe ), and ωe (respectively, ωe ) is the configuration obtained from ω by replacing ω(e) by 1 (respectively, 0). Let A be an increasing event, with indicator function 1 A . An edge e is called pivotal for the event A in the configuration ω if δe I A (ω) = 1. Show that the derivative of Pp ( A) equals the mean number of pivotal edges for A. Find a related formula for the second derivative of Pp ( A). What can you show for the third derivative, and so on? 4.10. [89] Show that every increasing subset of the cube [0, 1] N is Lebesguemeasurable. 4.11. Heads turn up with probability p on each of N coin flips. Let A be an increasing event, and suppose there exists a subgroup A of permutations of {1, 2, . . . , N } acting transitively, such that A is A-invariant. Let pc be the value of p such that P p ( A) = 21 . Show that there exists a constant c > 0 such that P p ( A) ≥ 1 − 21 N −c( p− pc ) ,

p ≥ pc ,

with a similar inequality for p ≤ pc . 4.12. Let µ be a positive measure on  = {0, 1} E satisfying the FKG lattice condition. For p ∈ (0, 1), let µ p be the probability measure given by   1 Y ω(e) 1−ω(e) µ(ω), p (1 − p) µ p (ω) = Zp e∈E

ω ∈ .

Let A be an increasing event. Show that µ p1 ( A)(1 − µ p2 ( A)) ≤ λ B( p2 − p1 ) ,

0 < p1 < p2 < 1,

where B=

inf

p∈( p1 , p2 )



 cξ p , p(1 − p)

and λ satisfies

  ξ p = min µ p (ω(e) = 1)µ p (ω(e) = 0) ,

2 max J A (e) ≤ λ, e∈E

c G. R. Grimmett 6 February 2009

e∈E

e ∈ E, p ∈ ( p1 , p2 ).

5 Further Percolation

The subcritical and supercritical phases of percolation are characterized respectively by the absence and presence of an infinite open cluster. Connection probabilities decay exponentially when p < pc , and there is a unique infinite cluster when p > pc . The power-law singularity at the phase transition is summarized. It is shown that pc = 12 for bond percolation on the square lattice. The Russo–Seymour–Welsh (RSW) method is described for site percolation on the triangular lattice, and this leads to a statement and proof of Cardy’s formula.

5.1 Subcritical phase In language borrowed from the theory of branching processes, a percolation process is termed subcritical if p < pc , and supercritical if p > pc . In the subcritical phase, all open clusters are (almost surely) finite. The chance of a long-range connection is small, and it approaches zero as the distance between the endpoints diverges. The process is considered to be ‘disordered’, and the probabilities of long-range connectivities tend to zero exponentially in the distance. Exponential decay may be proved by elementary means for sufficiently small p, as in the proof of Theorem 3.2, for example. It is quite another matter to prove exponential decay for all p < pc , and this was achieved for percolation by Aizenman and Barsky [6] and Menshikov [164, 165] around 1986. The principal result is the following therem, in which B(n) = [−n, n]d and ∂ B(n) = B(n) \ B(n − 1). (5.1) Theorem [6, 164]. There exists ψ( p), satisfying ψ( p) > 0 when 0 < p < pc , such that (5.2)

Pp (0 ↔ ∂ B(n)) ≤ e−nψ( p) ,

n ≥ 1.

The reader is referred to [95] for a full account of this important theorem. The two proofs of Aizenman–Barsky and Menshikov have some interesting similarities, while differing in fundamental ways. An outline of Menshikov’s proof c G. R. Grimmett 6 February 2009

[5.1]

Subcritical phase

73

is presented later in this section. The Aizenman–Barsky proof proceeds via an intermediate result, namely the following of Hammersley [114]. (5.3) Theorem [114]. Suppose that χ ( p) = E p |C| < ∞. There exists σ ( p) > 0 such that (5.4)

Pp (0 ↔ ∂ B(n)) ≤ e−nσ ( p) ,

n ≥ 1.

Seen in the light of Theorem 5.1, one may take the condition χ ( p) < ∞ as a characterization of the subcritical phase. It is not difficult to see, using subadditivity, that the limit of n −1 log Pp (0 ↔ ∂ B(n)) exists as n → ∞. See [95, Thm 6.10]. Proof. Let x ∈ ∂ B(n), and let τ p (0, x) = Pp (0 ↔ x) be the probability that there exists an open path of Ld joining the origin to x. Let N n be the number of vertices x ∈ ∂ B(n) with this property, so that the mean value of N n is (5.5)

E p (Nn ) =

X

τ p (0, x).

x∈∂ B(n)

Note that (5.6)

∞ X n=0

E p (Nn ) = =

∞ X X

τ p (0, x)

n=0 x∈∂ B(n)

X

τ p (0, x)

x∈Zd

= E p {x ∈ Zd : 0 ↔ x} = χ ( p).

If there exists an open path from the origin to some vertex of ∂ B(m + k), then there exists a vertex x in ∂ B(m) that is connected by disjoint open paths both to the origin and to a vertex on the surface of the translate ∂ B(k, x) = x + B(k), see Figure 5.1. By the BK inequality, (5.7)

Pp (0 ↔ ∂ B(m + k)) ≤ =

X

x∈∂ B(m)

X

x∈∂ B(m)

Pp (0 ↔ x)Pp (x ↔ x + ∂ B(k)) τ p (0, x)Pp (0 ↔ ∂ B(k))

by translation-invariance. Thus (5.8)

Pp (0 ↔ ∂ B(m + k)) ≤ E p (Nm )Pp (0 ↔ ∂ B(k)),

m, k ≥ 1.

The BK inequality makes this calculation simple, Hammersley [114] employed a more elaborate argument. c G. R. Grimmett 6 February 2009

74

[5.1]

Further Percolation

∂ B(m + k) ∂ B(m) x 0

Figure 5.1. The vertex x is joined by disjoint open paths to the origin and to the surface of the translate B(k, x) = x + B(k), indicated by the dashed lines.

P Let p be such that χ ( p) < ∞, so that ∞ m=0 E p (Nm ) < ∞ from (5.6). Then E p (Nm ) → 0 as m → ∞, and we may choose m such that η = E p (Nm ) satisfies η < 1. Let n be a positive integer and write n = mr + s where r and s are non-negative integers and 0 ≤ s < m. Then Pp (0 ↔ ∂ B(n)) ≤ Pp (0 ↔ ∂ B(mr )) ≤η

≤η

r

−1+n/m

since n ≥ mr

by iteration of (5.8) since n < m(r + 1),

which provides an exponentially decaying bound of the form of (5.4), valid for n ≥ m. It is left as an exercise to extend the inequality to n < m. 

Outline proof of Theorem 5.1. The full proof can be found in [94, 95, 165, 210]. Let S(n) be the ‘diamond’ S(n) = {x ∈ Zd : d(0, x) ≤ n} containing all points within graph-theoretic distance n of the origin, and write A n = {0 ↔ ∂ S(n)}. We are concerned with the probabilities g p (n) = Pp ( An ). By Russo’s formula, Theorem 4.75, (5.9)

g 0p (n) = E p (Nn )

where Nn is the number of pivotal edges for A n , that is, the number of edges e for which 1 A (ωe ) 6= 1 A (ωe ). By a simple calculation 1 1 (5.10) g 0p (n) = E p (Nn 1 An ) = E p (Nn | An )g p (n), p p c G. R. Grimmett 6 February 2009

[5.1]

Subcritical phase

75

which may be integrated to obtain (5.11)

 Z gα (n) = gβ (n) exp −  Z ≤ gβ (n) exp −

β

α

α

β

 1 E p (Nn | An ) d p p  E p (Nn | An ) d p ,

where 0 < α < β < 1. The vast majority of the work in the proof is devoted to showing that E p (Nn | An ) grows at least linearly in n when p < pc , and the conclusion of the theorem then follows immediately. The rough argument is as follows. Let p < pc , so that Pp ( An ) → 0 as n → ∞. In calculating E p (Nn | An ), we are conditioning on an event of diminishing probability, and thus it is feasible that there are many pivotal edges of A n . This will be proved by bounding (above) the mean distance between consecutive pivotal edges, and then applying a version of Wald’s equation. The BK inequality, Theorem 4.14, plays an important role. Suppose that An occurs, and denote by e1 , e2 , . . . , e N the pivotal edges for An , in the order in which they are encountered when building the cluster from the origin. It is easily seen that all open paths from the origin to ∂ S(n) traverse every e j . Furthermore, as illustrated in Figure 5.2, there must exist at least two edge-disjoint paths from the second endpoint of each e j (in the above ordering) to the first of e j +1 . Let M = max{k : Ak occurs}, so that Pp (M ≥ k) = g p (k) → 0

as k → ∞.

The key inequality states that (5.12)

Pp (Nn ≥ k | An ) ≥ P(M1 + M2 + · · · + Mk ≤ n − k),

where the Mi are independent copies of M. This is proved using the BK inequality, using the above observation concerning disjoint paths between consecutive pivotal edges. The proof is omitted here. By (5.12), (5.13)

Pp (Nn ≥ k | An ) ≥ P(M10 + M20 + · · · + Mk0 ≤ n),

where Mi0 = 1 + min{Mi , n}. Summing (5.13) over k, we obtain (5.14)

E p (Nn | An ) ≥ =

c G. R. Grimmett 6 February 2009

∞ X

k=1 ∞ X k=1

P(M10 + M20 + · · · + Mk0 ≤ n) Pp (K ≥ k + 1) = E(K ) − 1,

76

[5.1]

Further Percolation

∂ S(n)

e2 e3

e1 0

e4

Figure 5.2. Assume that 0 ↔ ∂ S(n). For any consecutive pair e j , e j +1 of pivotal edges, taken in the order of traversal from 0 to ∂ S(n), there must exist at least two edge-disjoint open paths joining the second vertex of e j and the first of e j +1 .

where K = min{k : M10 + M20 + · · · + Mk0 > n}. By Wald’s equation, n < E(S K ) = E(K )E(M10 ), whence E(K ) >

n n n = = Pn . 0 E(M1 ) 1 + E(min{M1 , n}) i =0 g p (i )

In summary, this shows that (5.15)

n − 1, i =0 g p (i )

E p (Nn | An ) ≥ Pn

0 < p < 1.

Inequality (5.15) may be fed into (5.10) to obtain a differential inequality for the g p (k). By a careful analysis of the latter inequality, one obtains that E p (Nn | An ) grows at least linearly with n whenever p satisfies 0 < p < pc . This step is neither short nor easy, but it is conceptually straightforward, and it completes the proof.  c G. R. Grimmett 6 February 2009

[5.2]

Supercritical phase

77

5.2 Supercritical phase The critical value pc is the value of p at which the percolation probability θ ( p) becomes strictly positive. It is widely believed that θ ( pc ) = 0, and this is perhaps the major conjecture of the subject. (5.16) Conjecture. For percolation on Ld with d ≥ 2, we have that θ ( pc ) = 0.

It is known that θ ( pc ) = 0 when either d = 2 (by results of [121], see Theorem 5.33) or d ≥ 19 (by the lace expansion of [118, 119]). The claim is believed to be canonical of percolation models on all lattices and in all dimensions. Suppose now that p > pc , so that θ ( p) > 0. What can be said about the number N of infinite open clusters? Since the event {N ≥ 1} is translation-invariant, it is trivial under the product measure Pp . However, Pp (N ≥ 1) ≥ θ ( p) > 0, whence Pp (N ≥ 1) = 1,

p > pc .

We shall see in the forthcoming Theorem 5.22 that Pp (N = 1) = 1 whenever θ ( p) > 0, which is to say that there exists a unique infinite open cluster throughout the supercritical phase. A supercritical percolation process in two dimensions may be studied in either of two ways. The first of these is by duality. Consider bond percolation on L 2 with density p. The dual process (as in the proof of the upper bound of Theorem 3.2) is bond percolation with density 1 − p. We shall see in Theorem 5.33 that the self-dual point p = 12 is also the critical point. Thus, the dual of a supercritical process is subcritical, and this enables a study of supercritical percolation on L 2 . A similar argument is valid for certain other lattices, although it is clear that the square lattice is special in that it is a self-dual graph. While duality is the technique for studying supercritical percolation in two dimensions, the process may also be studied by the block argument that follows. The block method was devised expressly for three and more dimensions in the hope that, amongst other things, it would imply the claim of Conjecture 5.16. Block arguments are a work-horse of the theory of general interacting systems. We assume henceforth that d ≥ 3 and that p is such that θ ( p) > 0; under this hypothesis, we wish to gain some control of the (a.s.) unique open cluster. The main result is the following, of which an outline proof is included later in the section. Let A ⊆ Zd , and write pc ( A) for the critical probability of bond percolation on the subgraph of Zd induced by A. Thus, for example, pc = pc (Zd ). Recall that B(k) = [−k, k]d . (5.17) Theorem [103]. Let d ≥ 3. If F is an infinite connected subset of Z d with pc (F) < 1, then for each η > 0 there exists an integer k such that pc (2k F + B(k)) ≤ pc + η. That is, for any set F sufficiently large that pc (F) < 1, one may ‘fatten’ F to a set having critical probability as close to pc as required. One particular application c G. R. Grimmett 6 February 2009

78

[5.2]

Further Percolation

of this theorem is to the limit of slab critical probabilities, and we elaborate on this next. Many results have been proved for subcritical percolation under the ‘finite susceptibility’ hypothesis that χ ( p) < ∞. The validity of this hypothesis for p < pc is implied by Theorem 5.1. Similarly, several important results for supercritical percolation have been proved under the hypothesis that ‘percolation occurs in slabs’. The two-dimensional slab Fk of thickness 2k is the set
Fk = Z2 × [−k, k]d−2 = Z2 × {0}d−2 + B(k),

with critical probability pc (Fk ). Since Fk ⊆ Fk+1 ⊆ Zd . the decreasing limit pc (F) = limk→∞ pc (Fk ) exists and satisfies pc (F) ≥ pc . The hypothesis of ‘percolation in slabs’ is that p > pc (F). By Theorem 5.17, (5.18)

lim pc (Fk ) = pc ,

k→∞

On of the best examples of the use of ‘slab percolation’ is the following estimate of the extent of a finite open cluster. (5.19) Theorem [60]. The limit   1 σ ( p) = lim − log Pp (0 ↔ ∂ B(n), |C| < ∞) n→∞ n exists. Furthermore σ ( p) > 0 if p > pc . This theorem asserts the exponential decay of a ‘truncated’ connectivity function when d ≥ 3. A similar result may be proved by duality for d = 2. We turn briefly to a discussion of the so-called ‘Wulff crystal’. Much attention has been paid to the sizes and shapes of clusters formed in models of statistical mechanics. When a cluster C is infinite with a strictly positive probability, but is constrained to have some large finite size n, then C is said to form a large ‘droplet’. The asymptotic shape of such a droplet as n → ∞ is prescribed in general terms by the theory of the so-called Wulff crystal, see the original paper [216] of Wulff. Specializing to percolation, we ask for properties of the open cluster C at the origin, conditioned on the event {|C| = n}. The study of the Wulff crystal is bound up with the law of the volume of a finite cluster. This has a tail that is ‘quenched exponential’: (5.20)

Pp (|C| = n) ≈ exp(−ρn (d−1)/d ),

where ρ = ρ( p) ∈ (0, ∞) for p > pc , and ≈ is to be interpreted in terms of exponential asymptotics. The explanation for the curious exponent is as follows. The ‘most economic’ way to create a large finite cluster is to find a region R containing a connected component D of size n, satisfying D ↔ ∞, and then c G. R. Grimmett 6 February 2009

[5.2]

Supercritical phase

79

Figure 5.3. Images of the Wulff crystal for the two-dimensional Ising model at two distinct temperatures, produced by simulation in time. The simulations were for finite time, and the images are therefore only approximations to the true crystals. The pictures are 1024 pixels square, and the inverse-temperatures are β = 34 , 10 11 . The corresponding random-cluster −4/3 −10/11 models have q = 2 and p = 1 − e ,1− e .

to cut all connections leaving R. Since p > pc , such regions R exist with |R| (respectively, |∂ R|) having order n (respectively, n (d−1)/d ), and the ‘cost’ of the construction is exponential in |∂ R|. The above argument yields a lower bound for Pp (|C| = n) of the quenchedexponential type, but considerably more work is required to show the exact asymptotic of (5.20), and indeed one obtains more. The (conditional) shape of Cn −1/d converges as n → ∞ to the solution of a certain variational problem, and the asymptotic region is termed the ‘Wulff crystal’ for the model. This is not too hard to make rigorous when d = 2, since the external boundary of C is then a closed curve. Serious technical difficulties arise when pursuing this programme when d ≥ 3. See [55] for an account and a bibliography. Outline proof of Theorem 5.19. The existence of the limit is an exercise in subadditivity of a standard type, although with some complications in this case (see [59, 95]). We sketch here a proof of the important estimate σ ( p) > 0. Let Sk be the (d − 1)-dimensional slab Sk = [0, k] × Zd−1 .

Since p > pc , we have by Theorem 5.17 that p > pc (Sk ) for some k, and we choose k accordingly. Let Hn be the hyperplane of vertices x of Ld with x 1 = n. It suffices to prove that (5.21)

Pp (0 ↔ Hn , |C| < ∞) ≤ e−γ n

for some γ = γ ( p) > 0. Define the slabs

Ti = {x ∈ Zd : (i − 1)k ≤ x 1 < i k},

c G. R. Grimmett 6 February 2009

1 ≤ i < bn/kc.

80

[5.2]

Further Percolation

H3k

0

T1

T2

T3

Figure 5.4. All paths from the origin to H3k traverse the regions Ti , i = 1, 2, 3.

Any path from 0 to Hn traverses each Ti . Since p > pc (Sk ), each slab contains (a.s.) an infinite open cluster. See Figure 5.4. If 0 ↔ Hn and |C| < ∞, then all paths from 0 to Hn must evade all such clusters. There are bn/kc slabs to traverse, and a price is paid for each. Modulo a touch of rigour, this implies that Pp (0 ↔ Hn , |C| < ∞) ≤ [1 − θk ( p)]bn/kc where

θk ( p) = Pp (0 ↔ ∞ in Sk ) > 0.

The inequality σ ( p) > 0 is proved.



Outline proof of Theorem 5.17. The full proof can be found in [95, 103]. For simplicity, we shall take F = Z2 ×{0}d−2 , so that 2k F + B(k) = Z2 ×[−k, k]d−2 . There are two main steps in the proof. In the first, we show the existence of long finite paths. In the second, we show how to take such finite paths and build an infinite cluster in a slab. The principal parts of the first step are as follows. Let p be such that θ ( p) > 0. 1. Let  > 0. Since θ ( p) > 0, there exists m such that Pp (B(m) ↔ ∞) > 1 − . [This holds since there exists, almost surely, an infinite open cluster.] 2. Let n ≥ 2m, say, and let k ≥ 1. We may choose n sufficiently large that, with probability at least 1 − 2, B(m) is joined to at least k points in ∂ B(n). c G. R. Grimmett 6 February 2009

[5.2]

Supercritical phase

81

Figure 5.5. An illustration of the event that the block centred at the origin is open. Each black square is a seed.

[If, for some k, this fails for unbounded n, then there exists N such that B(m) ↔ / Zd \ B(N ).] 3. By choosing k sufficiently large, we may ensure that, with probability at least 1 − 3, B(m) is joined to some point of ∂ B(n), which is itself connected to a copy of B(m), lying ‘on’ the surface ∂ B(n) and every edge of which is open. [We may choose k sufficiently large that there are many non-overlapping copies of B(m) in the correct positions, indeed sufficiently many that, with high probability, one is totally open.] 4. The open copy of B(m), constructed above, may be used as a ‘seed’ for iterating the above construction. When doing this, we shall need some control over where the seed is placed. It may be shown that every face of ∂ B(n) contains (with large probability) a point adjacent to some seed, and indeed many such points. See Figure 5.5. [There is sufficient symmetry to deduce this by the FKG inequality.] Above is the scheme for constructing long finite paths, and we turn to the second step. 5. This construction is now iterated. At each stage there is a certain (small) probability of failure. In order that there be a strictly positive probability of an infinite sequence of successes, we iterate ‘in two independent directions’. With care, one may show that the construction dominates a certain supercritical site percolation process on L2 . 6. We wish to deduce that an infinite sequence of successes entails an infinite open path of Ld within the corresponding slab. There are two difficulties with this. First, since we do not have total control of the positions of the seeds, the actual path in Ld may leave every slab. This may be overcome by a process of ‘steering’, in which, at each stage, we choose a seed in such c G. R. Grimmett 6 February 2009

82

[5.3]

Further Percolation

a position as to compensate for any earlier deviation in space. 7. A greater problem is that, in iterating the construction, we carry with us a mixture of ‘positive’ and ‘negative’ information (of the form that ‘certain paths exist’ and ‘others do not’). In combining events we cannot use the FKG inequality. The practical difficulty is that, although we may have an infinite sequence of successes, there will generally be breaks in any corresponding open route to ∞. This is overcome by sprinkling down a few more open edges, that is, by working at edge-density p + δ where δ > 0, rather than at density p. In conclusion, we find that, if θ ( p) > 0 and δ > 0, then there exists, with large probability, an infinite ( p+δ)-open path in a slice of the form Tk = Z2 ×[−k, k]d−2 for sufficiently large k. The claim of the theorem follows. There are many details to be considered in carrying out the above programme, and these are omitted here. 

5.3 Uniqueness of the infinite cluster The principal result of this section is the following: for any value of p for which θ ( p) > 0, there exists (a.s.) a unique infinite open cluster. Let N = N (ω) be the number of infinite open clusters. (5.22) Theorem [12]. If θ ( p) > 0, then Pp (N = 1) = 1.

A similar conclusion holds for more general probability measures. The two principal ingredients of the generalization are the translation-invariance of the measure, and the so-called ‘finite-energy property’, that states that, conditional on the states of all edges except e, say, the state of e is 0 (respectively, 1) with a strictly positive (conditional) probability. Proof. We follow [50]. The claim is trivial if p = 0, 1, and we assume henceforth that 0 < p < 1. Let S = S(n) be the ‘diamond’ S(n) = {x ∈ Zd : d(0, x) ≤ n}, and let E S be the set of edges of Ld joining pairs of vertices in S. We write N S (0) (respectively N S (1)) for the total number of infinite open clusters when all edges in E S are declared to be closed (respectively open). Finally, M S denotes the number of infinite open clusters that intersect S. The sample space  = {0, 1}E is a product space with a natural family of translations, and Pp is a product measure on . Since N is a translation-invariant function on , it is almost surely constant, which is to say that d

(5.23) there exists k = k( p) ∈ {0, 1, 2, . . . } ∪ {∞} such that Pp (N = k) = 1. Next we show that the k in (5.23) necessarily satisfies k ∈ {0, 1, ∞}. Suppose that (5.23) holds with k < ∞. Since every configuration on E S has a strictly c G. R. Grimmett 6 February 2009

[5.3]

Uniqueness of the infinite cluster

83

positive probability, it follows by the almost sure constantness of N that Pp (N S (0) = N S (1) = k) = 1. Now N S (0) = N S (1) if and only if S intersects at most one infinite open cluster (this is where we use the assumption that k < ∞), and therefore Pp (M S ≥ 2) = 0. Clearly, M S is non-decreasing in S = S(n), and M S(n) → N as n → ∞. Therefore, (5.24)

0 = Pp (M S(n) ≥ 2) → Pp (N ≥ 2),

which is to say that k ≤ 1. It remains to rule out the case k = ∞. Suppose that k = ∞. We will derive a contradiction by using a geometrical argument. We call a vertex x a trifurcation if: (a) x lies in an infinite open cluster, (b) there exist exactly three open edges incident to x, and (c) the deletion of x and its three incident open edges splits this infinite cluster into exactly three disjoint infinite clusters and no finite clusters; Let Tx be the event that x is a trifurcation. By translation-invariance, Pp (Tx ) is constant for all x, and therefore (5.25)

1 Ep |S(n)|

 X

x∈S(n)

1Tx



= Pp (T0 ).

It will be useful to know that the quantity Pp (T0 ) is strictly positive, and it is here that we use the assumed infinity of infinite clusters. Let M S (0) be the number of infinite open clusters that intersect S when all edges of E S are declared closed. Since M S (0) ≥ M S , by the remarks around (5.24), Pp (M S(n) (0) ≥ 3) ≥ Pp (M S(n) ≥ 3) → Pp (N ≥ 3) = 1

as n → ∞.

Therefore, there exists m such that Pp (M S(m) (0) ≥ 3) ≥ 12 , and we set S = S(m). Note that: (a) the event {M S (0) ≥ 3} is independent of the states of edges in E S , (b) if the event {M S (0) ≥ 3} occurs, there exist x, y, z ∈ ∂ S lying in distinct infinite open clusters of Ed \ E S . c G. R. Grimmett 6 February 2009

84

[5.3]

Further Percolation

x

y 0 z

Figure 5.6. Take a diamond S that intersects at least three distinct infinite open clusters, and then alter the configuration inside S in order to create a configuration in which 0 is a trifurcation.

Let ω ∈ {M S (0) ≥ 3}, and pick x = x(ω), y = y(ω), z = z(ω) according to (b). If there is more than one possible such triple, we pick such a triple according to some predetermined rule. It is a minor geometrical exercise (see Figure 5.6) to verify that there exist in E S three paths joining the origin to (respectively) x, y, and z, and that these paths may be chosen in such a way that: (i) the origin is the unique vertex common to any two of them, and (ii) each touches exactly one vertex lying in ∂ S. Let Jx,y,z be the event that all the edges in these paths are open, and that all other edges in E S are closed. Since S is finite,  R Pp (Jx,y,z | M S (0) ≥ 3) ≥ min{ p, 1 − p} > 0,

where R = |E S |. Now,

Pp (0 is a trifurcation) ≥ Pp (Jx,y,z | M S (0) ≥ 3)Pp (M S (0) ≥ 3)  R ≥ 21 min{ p, 1 − p} > 0,

which is to say that Pp (T0 ) > 0 in (5.25). It follows from (5.25) that the mean number of trifurcations inside S = S(n) grows in the manner of |S| as n → ∞. On the other hand, we shall see next that the number of trifurcations inside S can be no larger than the size of the boundary of S, and this provides the necessary contradiction. This final step must be performed properly (see [50, 95]), but the following rough argument is appealing and may be made rigorous. Select a trifurcation (t1 , say) of S, and choose some vertex y1 ∈ ∂ S such that t1 ↔ y1 in S. We now select a new trifurcation t2 ∈ S. It may c G. R. Grimmett 6 February 2009

[5.4]

Phase transition

85

be seen, using the definition of the term ‘trifurcation’, that there exists y 2 ∈ ∂ S such that y1 6= y2 and t2 ↔ y2 in S. We continue similarly, at each stage picking a new trifurcation tk ∈ S and a new vertex yk ∈ ∂ S. If there are τ trifurcations in S, then we obtain τ distinct vertices yk of ∂ S. Therefore |∂ S| ≥ τ . However, by the remarks above, E p (τ ) is comparable to S. This is a contradiction for large n, since |∂ S| grows in the manner of n d−1 and |S| grows in the manner of n d . 

5.4 Phase transition Macroscopic functions, such as the percolation probability θ ( p) = Pp (|C| = ∞) and the mean cluster size χ ( p) = E p |C|, have singularities at p = pc , and there is overwhelming evidence that these are of ‘power law’ type. A great deal of effort has been invested by physicists and mathematicians in understanding the nature of the percolation phase-transition. The picture is now fairly clear when d = 2, owing to the very significant progress in recent years in relating critical percolation to the stochastic (Schramm–)L¨owner curve SLE6 . There remain however substantial difficulties to be overcome before this chapter of percolation theory can be declared written, even when d = 2. The case of large d (currently, d ≥ 19) is also well understood, through work based on the so-called ‘lace expansion’. Most problems remain open in the obvious case d = 3, and ambitious and brave students are directed thus, with caution. The nature of the percolation singularity is supposed to be canonical, in that it is expected to have certain general features in common with phase transitions of other models of statistical mechanics. These features are sometimes referred to as ‘scaling theory’ and they relate to ‘critical exponents’. There are two sets of critical exponents, arising firstly in the limit as p → pc , and secondly in the limit over increasing distances when p = pc . We summarize the notation in Table 5.8. The asymptotic relation ≈ should be interpreted loosely (perhaps via logarithmic asymptotics1 ). The radius of C is defined by rad(C) = max{n : 0 ↔ ∂[−n, n]d }. The limit as p → pc should be interpreted in a manner appropriate for the function in question (for example, as p ↓ pc for θ ( p), but as p → pc for κ( p)).

There are eight critical exponents listed in Table 5.8, denoted α, β, γ , δ, ν, η, ρ, 1, but there is no general proof of the existence of any of these exponents for arbitrary d. In general, the eight critical exponents may be defined for phase transitions in a quite large family of physical systems. However, it is not believed 1 We

say that f (x) is logarithmically asymptotic to g(x) as x → 0 (respectively, x → ∞) if log f (x)/ log g(x) → 1. This is often written as f (x) ≈ g(x). c G. R. Grimmett 6 February 2009

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Function percolation probability

Behaviour

Exponent

θ( p) ≈ ( p − pc )β

β

χ f ( p) ≈ | p − pc |−γ

γ

κ 000 ( p) ≈ | p − pc |−1−α

α

θ( p) = Pp (|C| = ∞)

truncated mean cluster size χ f ( p) = E p (|C|; |C| < ∞) number of clusters per vertex

κ( p) = E p (|C|−1 )

cluster moments χkf ( p) = E p (|C|k ; |C| < ∞) correlation length

f ( p) χk+1

χkf ( p)

≈ | p − pc |−1 , k ≥ 1

1

ξ( p) ≈ | p − pc |−ν

ν

cluster volume

P pc (|C| = n) ≈ n −1−1/δ

δ

cluster radius

P pc rad(C) = n ≈ n −1−1/ρ

ρ

connectivity function

P pc (0 ↔ x) ≈ kxk2−d−η

η

ξ( p)




Table 5.8. Eight functions and their critical exponents.

that they are independent variables, but rather that they satisfy the scaling relations: 2 − α = γ + 2β = β(δ + 1), 1 = δβ,

γ = ν(2 − η),

and, when d is not too large, the hyperscaling relations: dρ = δ + 1,

2 − α = dν.

The upper critical dimension is the largest value dc such that the hyperscaling relations hold for d ≤ dc . It is believed that dc = 6 for percolation. There is no general proof of the validity of the scaling and hyperscaling relations, although quite a lot is known when d = 2 and for large d. In the context of percolation, there is an analytical rationale behind the scaling relations, namely the ‘scaling hypotheses’ that
Pp (|C| = n) ∼ n −σ f n/ξ( p)τ
Pp (0 ↔ x, |C| < ∞) ∼ kxk2−d−η g kxk/ξ( p)

c G. R. Grimmett 6 February 2009

[5.4]

Phase transition

87

in the double limit as p → pc , n → ∞, and for some constants σ , τ , η and functions f , g. Playing loose with rigorous mathematics, the scaling relations may be derived from these hypotheses. Similarly, the hyperscaling relations may be shown to be not too unreasonable, at least when d is not too large. For further discussion, see [95]. We note some further points. Universality. It is believed that the numerical values of critical exponents depend only on the value of d, and are independent of the particular percolation model. Two dimensions. When d = 2, perhaps α = − 23 , β =

5 36 ,

γ =

43 18 ,

δ=

91 5 ,...

See (5.41). Large dimension. When d is sufficiently large (actually, d ≥ dc ) it is believed that the critical exponents are the same as those for percolation on a tree (the ‘mean-field model’), namely δ = 2, γ = 1, ν = 12 , ρ = 21 , and so on (the other exponents are found to satisfy the scaling relations). Using the first hyperscaling relation, this supports the contention that dc = 6. Such statements are known to hold for d ≥ 19, see [118, 119] and the remarks later in this section. Open challenges include to prove: – the existence of critical exponents, – universality, – the scaling relations, – the conjectured values when d = 2, – the conjectured values when d ≥ 6. Progress towards these goals has been positive. For sufficiently large d, exact values are known for many exponents, namely the values from percolation on a regular tree. There has been remarkable progress in recent years when d = 2, inspired largely by work of Schramm [189], enacted by Smirnov [196], and confirmed by the programme pursued by Lawler, Schramm, and Werner to understand SLE curves. See Section 5.6. We close this section with some further remarks on the case of large d. The expression ‘mean-field’ permits several interpretations depending on context. A narrow interpretation of the term ‘mean-field theory’ for percolation involves trees rather than lattices. For percolation on a regular tree, it is quite easy to perform exact calculations of many quantities, including the numerical values of critical exponents. That is, δ = 2, γ = 1, ν = 12 , ρ = 21 , and other exponents are given according to the scaling relations, see [95]. Turning to percolation on Ld , it is known as remarked above that the critical exponents agree with those of a regular tree when d is sufficiently large. In fact, this is believed to hold if and only if d ≥ 6, but progress so far assumes that d ≥ 19. In the following theorem, we write f (x) ' g(x) if there exist positive c G. R. Grimmett 6 February 2009

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[5.5]

Further Percolation

constants c1 , c2 such that c1 f (x) ≤ g(x) ≤ c2 f (x) for all x close to a limiting value. (5.26) Theorem [119]. For d ≥ 19, θ ( p) ' ( p − pc )1

χ ( p) ' ( pc − p)−1 ξ( p) ' ( pc − p)

f ( p) χk+1

χkf ( p)

− 12

' ( pc − p)−2

as p ↓ pc , as p ↑ pc , as p ↑ pc , as p ↑ pc , for k ≥ 1.

Note the strong form of the asymptotic relation ', and the identification of the critical exponents β, γ , 1, ν. The proof of Theorem 5.26 centres on a property known as the ‘triangle condition’. Define (5.27)

T ( p) =

X

x,y∈Zd

Pp (0 ↔ x)Pp (x ↔ y)Pp (y ↔ 0),

and consider the triangle condition: T ( pc ) < ∞. The triangle condition was introduced by Aizenman and Newman [15], who showed that it implied that χ ( p) ' ( pc − p)−1 as p ↑ pc . Subsequently other authors showed that the triangle condition implied similar asymptotics for other quantities. It was Takashi Hara and Gordon Slade [118] who verified the triangle condition for large d, exploiting a technique known as the ‘lace expansion’.

5.5 Open paths in annuli There is a very useful technique for building open paths with certain geometry in two dimensions. It leads to a proof that the chance of an open circuit within an annulus [−3n, 3n]2 \ [−n, n]2 is at least f (δ), where δ is the chance of an open crossing of the square [−n, n]2 , and f is a strictly positive function. This result was useful in some of the original proofs concerning the critical probability of bond percolation on L2 (see [95, Sect. 11.7]), and has re-emerged more recently as central to estimates that permit the proof of the Cardy formula and conformal invariance. It is commonly named after Russo [186] and Seymour–Welsh [195]. The RSW lemma will be stated and proved in this section, and utilized in the next three. Since our application in the Section 5.7 will be to site percolation on the triangular lattice, we shall phrase the RSW lemma in that context. It is left to the reader to adapt and develop the arguments of this section for bond percolation on c G. R. Grimmett 6 February 2009

[5.5]

Open paths in annuli

Figure 5.9. The triangular lattice

89

and the (dual) hexagonal lattice .


the square lattice (see Exercise 5.5). The triangular lattice T is drawn in Figure 5.9, together with its dual hexagonal lattice H. RSW theory is presented in [95, Sect. 11.7] for the square lattice L2 and general bond-density p. We could follow the same route here for the triangular lattice, but for the sake of variation (and with an eye to the applications in Section 5.7) we shall restrict ourselves to the case p = 21 and shall give a shortened proof due to Stanislav Smirnov. The more conventional approach may be found in [211], see also [210], and [42] for a variant on the square lattice. Thus, in this section we restrict ourselves to site percolation on T with density 21 . Each site of T is coloured black with probability 21 , and white otherwise, and the relevant probability measure is denoted as P. 2 2 The triangular lattice is embedded √ in R with vertex-set {mi+nj : (m, n) ∈ Z } 1 where i = (1, 0) and j = 2 (1, 3). Write Ra,b for the subgraph induced by vertices in the rectangle [0, a] × [0, b], √ and we shall restrict ourselves always to integers a and integer multiples b of 12 3. Let Ha,b be the event that there exists a black path that traverses Ra,b from its left side to its right side. The ‘engine room’ of the RSW method is the following lemma. (5.28) Lemma. P(H2a,b ) ≥ 14 P(Ha,b )2 . By iteration,

(5.29)

P(H2k a,b ) ≥ 4

1

4 P(Ha,b )

 2k

,

k ≥ 0.

As ‘input’ to this inequality, we prove the following. (5.30) Lemma. We have that P(Ha,a √3 ) ≥ 12 .

Let 3m be the set of vertices in T at graph-theoretic distance m or less from the origin 0, and define the annulus A n = 33n \ 3n−1 . Let On be the event that An contains a black circuit C such that 0 lies in the bounded component of R 2 \ C. c G. R. Grimmett 6 February 2009

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Further Percolation

b

g

ρg

Ug

0

ρUg

Jg 0

a

ρ Jg 2a

Figure 5.10. The crossing g and its reflection ρg in the box R 2a,b . The events Bg and Wρg are illustrated by the two lower paths, and exactly one of these events occurs.

(5.31) Theorem (RSW). There exists σ > 0 such that P(On ) > σ for all n ≥ 1.

Proof of Lemma 5.28. We follow an unpublished argument of Stanislav Smirnov 2 . Let g be a path that traverses Ra,b from left to right. Let ρ denote reflection in the line x = a, so that ρg connects the left and right sides of [a, 2a] × [0, b]. See Figure 5.10. Assume for the moment that g does not intersect the x-axis, and let Ug be the connected subgraph of Ra,b lying on or ‘beneath’ g. Let J g (respectively, Jb ) be the part of the boundary ∂U g (respectively, Ra,b ) lying on either the x-axis or y-axis, and let ρ J g (respectively, ρ Jb ) be its reflection. Let Bg be the event that there exists a black path of U g ∪ ρUg joining some vertex of g to some vertex of ρ J g . If Bg does not occur, then the event Wρg , that there exists a white path of U g ∪ ρUg from ρg to Jg , must occur. The events B g , Wρg are mutually exclusive with the same probability, and therefore P(Bg ) = P(Wρg ) = 12 .

(5.32)

The same relation holds if g touches the x-axis, with J g suitably adapted. Let L be the left side of R2a,b and R its right side. By the FKG inequality, P(H2a,b ) ≥ P(L ↔ ρ Jb , R ↔ Jb )

≥ P(L ↔ ρ Jb )P(R ↔ Jb ) = P(L ↔ ρ Jb )2 ,

where ↔ denotes connection by a black path. Let γ be the ‘highest’ black path from the left to the right sides of Ra,b , if such a path exists. Conditional on the event {γ = g}, the states of sites beneath g are 2 See

also [209].

c G. R. Grimmett 6 February 2009

[5.6]

Open paths in annuli

91

Figure 5.11. If each of six long rectangles are traversed in the long direction by black paths, then the intersection of these paths contains a black cycle within the annulus A n .

independent Bernoulli variables, whence, in particular, the events B g and {γ = g} are independent. By (5.32), X X P(L ↔ ρ Jb ) ≥ P(γ = g, Bg ) = P(Bg )P(γ = g) g

= and the lemma is proved.

1 2

X g

g

P(γ = g) = 21 P(Ha,b ),



Proof of Lemma 5.30. This is similar to the argument leading to (5.32). Consider the rhombus R of T comprising all vertices of the form mi + nj for 0 ≤ m, n ≤ 2a. Let B be the event that R is traversed from left to right by a black path, and W the event that it is traversed from top to bottom by a white path. These two events are mutually exclusive with the same probability, and one or the other necessarily occurs. Therefore P(B) = 21 . On B, there exists a left–right crossing of the √  (sub-)rectangle [a, 2a] × [0, a 3], and the claim follows. Proof of Theorem 5.31. By (5.29) and Lemma 5.30, there exists α > 0 such that P(H8n,n √3 ) ≥ α,

n ≥ 1.

We may represent the annulus A n as the pairwise-intersection of six copies of R8n,n √3 obtained by translation and rotation, as illustrated in Figure 5.11. If each of these is traversed by a black path in its long direction, than the event O n occurs. By the FKG inequality, P(On ) ≥ α 6 ,

and the theorem is proved.

c G. R. Grimmett 6 February 2009



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[5.6]

Further Percolation

5.6 The critical probability in two dimensions We revert to bond percolation on the square lattice in this section. The square lattice has a property of self-duality, illustrated in Figure 3.1. ‘Percolation of open edges on the primal lattice’ is dual to ‘percolation of closed edges on the dual lattice’. The self-dual value of p is thus p = 21 , and it was long been believed by physicists that the self-dual point is also the critical point pc . Theodore Harris [121] proved by a geometric construction that θ ( 21 ) = 0, whence pc (Z2 ) ≥ 12 . Harry Kesten [135] proved the complementary inequality. (5.33) Theorem [121, 135]. The critical probability of bond percolation on the square lattice equals 21 . Furthermore, θ ( 21 ) = 0. Before giving a proof, we make some comments on the original proof. Harris [121] showed that, if θ ( 21 ) > 0, then one can construct closed dual circuits around the origin. Such circuits prevent the cluster C from being infinite, and therefore θ ( 12 ) = 0, a contradiction. Similar ‘path-construction’ arguments were developed by Russo [186] and Seymour–Welsh [195] in a proof that p > p c if and only if χ (1 − p) < ∞. This so-called ‘RSW method’ has acquired prominence in recent work on SLE (see Sections 5.5 and 5.7). The complementary inequality pc (Z2 ) ≤ 12 was proved by Kesten in [135]. More specifically, he showed that, for p < 21 , the probability of an open left– right crossing of the rectangle [0, 2k ] × [0, 2k+1 ] tends to zero as k → ∞. With the benefit of hindsight, one may view his argument as establishing a type of sharp-threshold theorem for the event in question. The arguments that prove Theorem 5.33 may be adapted to certain other situations. For example, Wierman [211] has proved similarly that the critical probabilities of bond percolation on the hexagonal/triangular pair of lattices (see Figure 5.9) are the dual pair of values satisfying the star–triangle transformation. Russo [187] adapted the arguments to site percolation on the square lattice. It is easily seen by the same arguments3 that site percolation on the triangular lattice has critical probability 21 . The proof of Theorem 5.33 is broken into two parts. Proof of Theorem 5.33: θ ( 21 ) = 0, and hence pc ≥ 21 . Zhang discovered a beautiful proof of this, using only the uniqueness of the infinite cluster, see [95]. Set p = 21 . Let T (n) = [0, n]2 , and find N sufficiently large that P 1 (∂ T (n) ↔ ∞) > 1 − ( 18 )4 , 2

n ≥ N.

We set n = N + 1. Let Al , Ar , At , Ab be the (respective) events that the left, right, 3 See

also Section 5.8.

c G. R. Grimmett 6 February 2009

[5.6]

The critical probability in two dimensions

93

T (n)

Figure 5.12. The left and right sides of the box T (n) are joined to infinity by open paths of the primal lattice, and the top and bottom sides of the dual box T (n) d are joined to infinity by closed dual paths. Using the uniqueness of the infinite open cluster, the two open paths must be joined by an open path. This forces the existence of two disjoint infinite closed clusters in the dual.

top, bottom sides of T (n) are joined to ∞ off T (n). By the FKG inequality, P 1 (T (n) ↔ / ∞) = P 1 ( Al ∩ Ar ∩ At ∩ Ab ) 2

2

≥ P 1 ( Al )P( Ar )P( At )P( Ab ) 2

= P 1 ( A g )4 2

by symmetry, for g = l, r, t, b. Therefore

 1/4 P 1 ( Ag ) ≥ 1 − 1 − P 1 (T (n) ↔ ∞) > 78 . 2

2

We consider next the dual box, with vertex set T (n)d = [0, n − 1]2 + ( 21 , 21 ). Let Ald , Ard , Atd , Abd denote the (respective) events that the left, right, top, bottom sides of T (n)d are joined to ∞ by a closed dual path off T (n)d . Since each edge of the dual is closed with probability 21 , g

P 1 ( Ad ) > 87 , 2

g = l, r, t, b.

Consider the event A = A l ∩ Ar ∩ Atd ∩ Abd , illustrated in Figure 5.12. Clearly, P 1 ( A) ≤ 21 , so that P 1 ( A) ≥ 21 . However, on A, either L2 has two infinite 2

2

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Further Percolation

Figure 5.13. If there is no open left–right crossing of S(n),there must exist a closed top–bottom crossing in the dual.

open clusters, or its dual has two infinite closed clusters. By Theorem 5.22, each event has probability 0, a contradiction. We deduce that θ ( 12 ) = 0, implying in particular that pc ≥ 21 .  Proof of Theorem 5.33: pc ≤ 21 . We give two proofs. The first uses the general exponential-decay Theorem 5.1. The second is previously unpublished work of Stanislav Smirnov, and is close in spirit to Kesten’s original proof. Proof A. Suppose instead that pc > 21 . By Theorem 5.1, there exists γ > 0 such that P 1 (0 ↔ ∂[−n, n]2 ) ≤ e−γ n ,

(5.34)

2

n ≥ 1.

Let S(n) be the graph with vertex set [0, n + 1] × [0, n] and edge set containing all edges inherited from L2 except those in either the left side or the right side of S(n). Let A be the event that there exists an open path joining the left side and right side of S(n). If A does not occur, then the top side of the dual of S(n) is joined to the bottom side by a closed dual path. Since the dual of S(n) is isomorphic to S(n), and since p = 12 , it follows that P 1 ( A) = 21 . See Figure 5.13. However, by 2 (5.34), P 1 ( A) ≤ (n + 1)e−γ n , 2

a contradiction for large n. We deduce that pc ≤ 12 .

Proof B. Let 3r = [−r, r ]2 , and Ar = 33r \ 3r be an ‘annulus’. The principal ingredient is an estimate that follows from the square-lattice version 4 of the RSW 4 See

Exercise 5.5.

c G. R. Grimmett 6 February 2009

[5.6]

The critical probability in two dimensions

95

Theorem 5.31. There exist c 0 log r disjoint annuli 3k within [−r, r ]2 , and each of these contains a dual closed circuit with probability at least σ > 0. Therefore, g(r ) = P 1 (0 ↔ ∂3r ) satisfies 2

(5.35)

g(r ) ≤ (1 − σ )c

0 log r

= r −α ,

where c0 , α > 0. There is a variety of ways of implementing the basic argument of this proof, of which we choose the following. Let Rn = [0, 2n] × [0, n] where n ≥ 1, and let Hn be the event that Rn is traversed by an open path from left to right. In the above notation, P 1 ( A) = 21 , and hence, by Lemma 5.28 rewritten for the square 2 lattice, there exists γ > 0 such that (5.36)

P 1 (Hn ) ≥ γ , 2

n ≥ 1.

We take p ≥ 12 and work with the dual model. Let Sn be the dual box ( 12 , 12 ) + [0, 2n − 1] × [0, n + 1], and let Vn be the event that Sn is traversed from top to bottom by a closed dual path. Let N n be the number of pivotal edges for the event Vn , and let 5 be the event that N n ≥ 1 and all pivotal edges are closed (in the dual). We claim that (5.37)


Pp (Nn = k − 1 | 5) ≤ kg (n − k)/k ,

1 ≤ k < 12 n,

of which the proof follows. For any top–bottom path l of Vn , we write L(l) (respectively, R(l) for the set of edges of Sn lying to the ‘left’ (respectively, ‘right’) of l. On 5, there exists a closed top–bottom path of Sn , and from amongst such paths we may pick the leftmost, denoted λ. As in the proof of Lemma 5.28, λ is measurable on the states of edges in and to the left of λ; that is to say, for any admissible l, the event {λ = l} depends only on the states of edges in l ∪ L(l). Suppose that λ = l, so that every pivotal edge for Vn necessarily lies in l. We now take a walk along l from bottom to top, encountering in order the edges of l. Let f 1 , f 2 , . . . , f k be the pivotal edges thus encountered, with f i = hxi , yi i, and let y0 be the initial vertex of l and x k+1 the last. For i = 0, 1, . . . , k, there exists a closed path of R(l) from yi to xi +1 . See Figure 5.14. Suppose Nn = k − 1 for some k < 21 n. As we move along l, we progressively reveal the closed clusters C i of the yi . That is, we first observe C 0 , a cluster that contains an open path of R(l) joining y0 to x 1 . Then we observe C 1 , a cluster containing a path of R(l) from y1 to x 2 , and so on. There exists j such that the L ∞ -distance between y j and x j +1 is at least (n − k)/k. Therefore, in this sequence of observations, there exists j such that C j contains a path of R(l) from y j to y j + ∂3(n−k)/k . Conditional on the history of the process up to step j , the chance of this is no greater than g((n − k)/k). Inequality (5.37) follows. c G. R. Grimmett 6 February 2009

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Further Percolation

x4

l f3

L(l)

f2 R(l) f1 y0 Figure 5.14. Between any two successive pivotal edges of the top–bottom crossing l, there exists a closed path joining their endpoints and (otherwise) lying entirely in R(l). There are three pivotal edges f i in this illustration, and the dashed lines are the closed connections of R(l) joining successive f i .

By (5.37) and (5.35), E p (Nn | 5) ≥ k · Pp (Nn ≥ k | 5)   α  k 2 ≥k 1−k n−k We choose k = 12 n β where 0 < β < α/(2 + α), to obtain that (5.38)

E p (Nn | 5) ≥ cn β ,

n ≥ 1,

where c > 0 is an absolute constant. We prove next that (5.39)

Pp (5) ≥ Pp (Hn )Pp (Vn ).

Suppose Vn occurs, with λ = l, and let Wl be the event that there exists e ∈ l with the following property: the dual edge e d = hu, vi has an endpoint, v say, that is joined to the right side of Rn by an open primal path of edges dual to edges in R(l). By the definition of leftmost crossing, it is automatically the case that the other endpoint u is joined to the left side of Rn by a primal open path of edges dual to members of L(l). Since Pp (Wl | λ = l) ≥ Pp (Hn ), Pp (Hn )Pp (Vn ) ≤

X

c G. R. Grimmett 6 February 2009

l

Pp (λ = l)Pp (Wl | λ = l) = Pp (5).

[5.6]

The critical probability in two dimensions

97

Figure 5.15. The boxes with aspect ratio 2 are arranged in such a way that, if all but finitely many are traversed in the long direction, then there must exist an infinite cluster

Since Pp (Hn ) = 1 − Pp (Vn ), by (5.38) and Russo’s formula (Theorem 4.75), d Pp (Hn ) ≥ E p (Nn ) ≥ cn β Pp (Hn )[1 − Pp (Hn )], dp

p ≥ 12 .

The resulting differential inequality   1 d 1 − Pp (Hn ) ≥ cn β Pp (Hn ) 1 − Pp (Hn ) d p may be integrated over the interval [ 21 , p] to obtain5 via (5.36) that 1 − Pp (Hn ) ≤ from which we extract the fact that ∞ X

(5.40)

n=1

 1 exp −c( p − 21 )n β . γ

[1 − Pp (Hn )] < ∞,

p > 21 .

We now use a block argument 6 that was published in [58]. Consider the nested rectangles B2r −1 = [0, 22r ] × [0, 22r −1 ],

B2r = [0, 22r ] × [0, 22r +1 ],

r ≥ 1,

illustrated in Figure 5.15. Let K 2r −1 (respectively, K 2r ) be the event that B2r −1 (respectively, B2r ) is traversed from left to right (respectively, top to bottom) by an open path, so that Pp (K k ) = Pp (H2k ). By (5.40) and the Borel–Cantelli lemma, all but finitely many of the K k occur, Pp -almost surely. By Figure 5.15 again, this entails the existence of an infinite open cluster, whence θ ( p) > 0 and p c ≤ 12 .  5 The 6 An

same point may be reached using the theory of influence, as in Exercise 5.4. alternative block argument may be found in Section 5.8.

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[5.7]

Further Percolation

5.7 Cardy formula There is a rich physical theory of phase transitions in theoretical physics, and critical percolation is at the heart of this theory. The case of two dimensions is very special, in that methods of conformality and complex analysis, linked to predictions of conformal field theory, have given rise to a beautiful and universal vision for the nature of such singularities. This vision is both analytical and geometrical. Its proof has been one of the principal targets of probability theory and theoretical physics over recent decades. The “road-map” to the proof is now widely accepted, and many key ingredients have become clear. There remain some significant problems. The principal ingredient of the theory is the SLE process introduced in Section 2.5. In a classical theorem of L¨owner [155], one sees that a growing curve γ in R2 may be encoded via conformal maps gt in terms of a so-called ‘driving function’ b : [0, ∞) → R. Oded Schramm [189] predicted that a variety of scaling limits of stochastic process in R2 may be formulated thus, with b chosen as a Brownian motion with an appropriate variance parameter κ. He gave a partial proof that LERW on Z2 , suitably re-scaled, has limit SLE2 , and he indicated that UST has limit SLE8 and percolation SLE6 . These observations did not come out of the blue. There was considerable earlier speculation around the idea of conformality, and we highlight the statement by John Cardy of his formula [54], and the discussions of Michael Aizenman and others concerning possible invariance under conformal maps (see, for example, [4, 5, 141]). Much has been achieved since Schramm’s paper [189]. Stanislav Smirnov [196, 197] has proved that critical site percolation on the triangular lattice satisfies Cardy’s formula, and his route to ‘complete conformality’ and SLE 6 has been verified, see [51, 52] and [209]. Many of the critical exponents for the model have now been calculated rigorously, namely (5.41)

β=

43 4 48 18 , ν = 3 , ρ = 5 , exponent 54 , see [146, 200]. On

5 36 ,

γ =

the other hand, it has together with the ‘two-arm’ not yet been possible to extend such results to other principal percolation models such as bond or site percolation on the square lattice (some extensions have proved possible, see [61] for example). On a related front, Smirnov [198, 199] has proved convergence of the re-scaled cluster boundaries of the critical Ising model (respectively, the associated randomcluster model) on Z2 to SLE3 (respectively, SLE16/3 ). This will be extended in [62] to the Ising model on any so-called isoradial graph, that is, a graph embeddable in R2 in such a way that the vertices of any face lie on the circumference of some circle of given radius r . The theory of SLE will soon constitute a book in its own right 7 , and similarly for the theory of the several scaling limits that have now been proved. These 7 See

[143].

c G. R. Grimmett 6 February 2009

[5.7]

Cardy formula

x

99

C = eπi /3

b D

T

φ

X

a c

A=0

B=1

Figure 5.16. The conformal map φ is a bijection from D to the interior of T , and extends uniquely to the boundaries.

general topics are beyond the scope of the current work. We restrict ourselves here to the statement and proof of Cardy’s formula for critical site percolation on the triangular lattice, and we make use of the accounts to be found in [209, 210]. See also [25, 44, 180]. We consider site percolation on the triangular lattice T, with density p = 21 of open (or ‘black’) vertices. It may be proved very much as in Theorem 5.33 that pc = 12 for this process, but this fact does not appear to be directly relevant to the material that follows. It is, rather, the ‘self-duality’ or ‘self-matching’ property that counts. Let D (6= C) be an open simply connected domain in R2 ; for simplicity we shall assume that its boundary ∂ D is a Jordan curve. Let a, b, c be distinct points of ∂ D, taken in anticlockwise order around ∂ D. There exists a conformal map φ from D to the interior of the equilateral triangle T of C with vertices A = 0, B = 1, C = eπi /3 , and such φ can be extended to the boundary ∂ D in such a way that it becomes a homeomorphism from D ∪ ∂ D (respectively, ∂ D) to the closed triangle T (respectively, ∂ T ). There exists a unique such φ that maps a, b, c to A, B, C, respectively. With φ chosen accordingly, the image X = φ(x) of a fourth point x ∈ ∂ D, taken for example on the arc from b to c, lies on the arc BC of T . See Figure 5.16. The triangular lattice T is re-scaled to have mesh-size δ, and we ask about the probability Pδ (ac ↔ bx in D) of an open path joining the arc ac to the arc bx, in an approximation to the intersection (δ T) ∩ D of the re-scaled lattice with D. It is a standard application of the RSW method of the last section to show that Pδ (ac ↔ bx in D) is uniformly bounded away from 0 and 1 as δ → 0. It thus seems reasonable that this probability should converge as δ → 0, and Cardy’s formula (together with conformality) tells us the value of the limit. (5.42) Theorem. Cardy formula [54, 196, 197]. In the notation introduced above, (5.43)

Pδ (ac ↔ bx in D) → |B X |

as δ → 0.

Some history: In [54], John Cardy stated the limit of Pδ (ac ↔ bx in D) as a hypergeometric function of a certain cross-ratio. His derivation was based on c G. R. Grimmett 6 February 2009

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C = Aτ 2

z

A = A1

B = Aτ

Figure 5.17. An illustration of the event E 1n (z), that z is separated from A τ Aτ 2 by a black

path joining A 1 Aτ and A1 Aτ . 2

arguments from conformal field theory, and was widely accepted as an inspired piece of physics. Lennart Carleson recognised the hypergeometric function in terms of the conformal map from a rectangle to a triangle, and was led to conjecture the simple form of (5.43). The limit was proved in 2001 by Stanislav Smirnov [196, 197]. The proof utilizes the three-way symmetry of the triangular lattice in a somewhat mysterious way. The Cardy formula is, in a sense, only the beginning of the story of the scaling limit of critical two-dimensional percolation. It leads naturally to a full picture of the scaling limits of open paths, within the context of the Schramm–L o¨ wner evolution SLE6 . One explicit application is towards the calculation of critical exponents [146, 200], but SLE6 presents a much fuller picture than this. Further details may be found in [52, 53, 197, 209]. The principal open problem at the time of writing is to extend the scaling limit beyond site triangular model to either the bond or site model on another major lattice. We prove Theorem 5.42 in the remainder of this section. This will be done first with D = T , the unit equilateral triangle, followed by the general case. Assume then that D = T with T given as above. The vertices of T are A = 0, B = 1, C = ei π/3 . We take δ = 1/n, and shall later let n → ∞. Consider site percolation on Tn = (n −1 T) ∩ T . One may draw either Tn or its dual graph Hn , which comprises hexagons with centres at the vertices of Tn . Each vertex of Tn (or equivalently, each face of Hn ) is declared black with probability 12 , and white otherwise. For ease of notation later, we write A = A 1 , B = Aτ , C = Aτ 2 , where τ = e2πi /3 .

For vertices V , W of T we write V W for the arc of the boundary of T from V to W. c G. R. Grimmett 6 February 2009

[5.7]

Cardy formula

101

Let z be the centre of a face of Tn (or equivalently, z ∈ V (Hn ), the vertex-set of the dual graph Hn ). The events to be studied are as follows. Let E 1n (z) be the event that there exists a self-avoiding black path from A 1 Aτ to A1 Aτ 2 that separates z from Aτ Aτ 2 . Let E τn (z), E τn2 (z) be given similarly after rotating the triangle clockwise by τ and τ 2 , respectively. The event E 1n (z) is illustrated in Figure 5.17. We write j = 1, τ, τ 2 .

Hjn (z) = P(E jn (z)),

(5.44) Lemma. The functions Hjn , j = 1, τ, τ 2 , are uniformly H¨older on V (Hn ), in that there exist absolute constants c ∈ (0, ∞),  ∈ (0, 1) such that (5.45) (5.46)

|Hjn (z) − Hjn (z 0 )| ≤ c|z − z 0 | ,

1 − Hjn (z) ≤ c|z − A j | ,

z, z 0 ∈ V (Hn ), z ∈ V (Hn ).

where A j is interpreted as the complex number at the vertex A j . The domain of the Hjn may be extended as follows: the set V (Hn ) may be viewed as the vertex-set of a triangulation of a region slightly smaller than T , on each triangle of which Hjn may be defined by linear interpolation between its values at the three vertices. Finally, the Hjn may be extended up to the boundary of T in such a way that the resulting functions satisfy (5.45) for all z, z 0 ∈ T , and (5.47)

Hj ( A j ) = 1,

j = 1, τ, τ 2 .

1 , Proof. It suffices to prove (5.45) for small |z − z 0 |. Suppose that |z − z 0 | ≤ 100 say, and let F be the event that there exist both a black and a white circuit of the entire re-scaled triangular lattice T/n, each of diameter smaller that 41 , and each having both z and z 0 in the bounded component of its complement. If F occurs, then either both or neither of the events E jn (z), E jn (z 0 ) occur, whence

|Hjn (z) − Hjn (z 0 )| ≤ 1 − P(F). When z and z 0 are a ‘reasonable’ distance from A j , the white circuit prevents the occurrence of one of these events without the other. The black circuit is needed when z, z 0 are close to A j . There exists C > 0 such that one may find log(C/|z − z 0 |) vertex-disjoint annuli, each containing z, z 0 in their central ‘hole’, and each within distance 18 of both z and z 0 (the definition of annulus precedes Theorem 5.31). By Theorem 5.31, the chance that no such annulus contains a black (respectively, white) circuit is no greater than 0 (1 − σ )log(C/|z−z |) whence 1 − P(F) ≤ c|z − z 0 | for suitable c and . Inequality (5.46) follows similarly with z 0 = A j .  c G. R. Grimmett 6 February 2009

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Further Percolation

C = Aτ 2

l1 s1 z3

z2 z

l2 s2

s3 z1

l3 B = Aτ

A = A1

Figure 5.18. An illustration of the event E 1n (z1 ) \ E 1n (z). The path l 1 is white, and l 2 , l3 are black.

It is convenient to work in the space of uniformly Ho¨ lder functions on the closed triangle T satisfying (5.45)–(5.46). By the Arzel`a–Ascoli theorem (see, for example, [67, Sect. 2.4]), this space is relatively compact. Therefore, the sequence of triples (H1n , Hτn , Hτn2 ) possesses subsequential limits in the sense of uniform convergence, and we shall see that any such limit is of the form (H1 , Hτ , Hτ 2 ) where the Hj are harmonic with certain boundary conditions, and satisfy (5.45)– (5.46). The boundary conditions guarantee the uniqueness of the H j , and it will follow that Hjn → Hj as n → ∞. We shall see in particular that

Hτ 2 (z) =

√2 |=(z)|, 3

the normalized imaginary part of z. The values of H1 and Hτ are found by rotation. The claim of the theorem will follow by letting z → X ∈ BC. Let (H1 , Hτ , Hτ 2 ) be a subsequential limit as above. That the H j are harmonic will follow from the fact that the functions (5.48)

G 1 = H1 + Hτ + Hτ 2 ,

G 2 = H1 + τ Hτ + τ 2 Hτ 2 ,

are analytic, and this analyticity will be implied by Morera’s theorem on checking that the contour integrals of G 1 , G 2 around triangles of a certain form are zero. The integration step amounts to summing the H j (z) over certain z and using a cancellation property that follows from the next lemma. Let z be the centre of a face of Tn , and let z 1 , z 2 , z 3 be the centres of the neighbouring faces ordered anticlockwise around z. See Figure 5.18. c G. R. Grimmett 6 February 2009

[5.7]

Cardy formula

103

C = Aτ 2

A = A1

B = Aτ

Figure 5.19. The exploration path ηn started at the top vertex A τ 2 and stopped when it hits the bottom side A 1 Aτ of the triangle.

(5.49) Lemma. We have that P[E 1n (z 1 ) \ E 1n (z)] = P[E τn (z 2 ) \ E τn (z)] = P[E τn2 (z 3 ) \ E τn2 (z)]. Before proving this, we introduce the exploration process illustrated in Figure 5.19. Suppose that all vertices ‘just outside’ the arc A 1 Aτ 2 (respectively, Aτ Aτ 2 ) of Tn are black (respectively, white). The exploration path is defined to be the unique path ηn on the edges of the dual (hexagonal) graph, beginning immediately above Aτ 2 and descending to A 1 Aτ such that: as one traverses ηn from top to bottom, the vertex immediately on one’s left (respectively, right), looking along the path from Aτ 2 , is white (respectively, black). When traversing ηn thus, there is a white path on one’s left and a black path on one’s right. Proof. The event E 1n (z 1 ) \ E 1n (z) occurs if and only if there exist disjoint paths l1 , l2 , l3 of Tn such that: (i) l1 is white and joins s1 to Aτ Aτ 2 , (ii) l2 is black and joins s2 to A1 Aτ 2 , (iii) l3 is black and joins s3 to A1 Aτ . See Figure 5.18 for an explanation of the notation. On this event, the exploration path ηn of Figure 5.19 passes through z and arrives at z along the edge hz 3 , zi of Hn . Furthermore, up to the time at which it hits z, it lies in the region of Hn between l2 and l1 . Indeed we may take l 2 (respectively, l1 ) to be the maximal black path (respectively, white path) of Tn lying on the right side (respectively, left side) of ηn up to this point. Conditional on the event above, and with l 1 and l2 given in terms of ηn accordingly, the states of vertices of Tn lying below l1 ∪ l2 are independent Bernoulli c G. R. Grimmett 6 February 2009

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[5.7]

Further Percolation

variables. Thus the conditional probability of a black path l 3 satisfying (iii) is the same as that of a white path. We make this measure-preserving change, and then we interchange the colours white/black to conclude that: E 1n (z 1 ) \ E 1n (z) has the same probability as the event that there exist disjoint paths l 1 , l2 , l3 of Tn such that: (i) l1 is black and joins s1 to Aτ Aτ 2 , (ii) l2 is white and joins s2 to A1 Aτ 2 , (iii) l3 is black and joins s3 to A1 Aτ . This is precisely the event E τn (z 2 ) \ E τn (z), and the lemma is proved. 

We use Morera’s theorem in order to show the necessary analyticity. This Htheorem states that: if f : R → C is continuous on the open region R, and γ f dz = 0 for all closed curves γ in R, then f is analytic. It is standard (see [185, p. 208]) that it suffices to consider triangles γ in R. One may in fact restrict oneself to equilateral triangles with one side parallel to the x-axis. This may be seen either by an approximation argument, or by an argument based on the threefold Cauchy–Riemann equations 1 ∂f 1 ∂f ∂f = = 2 2, ∂1 τ ∂τ τ ∂τ

(5.50)

where ∂/∂ j means the derivative in the direction of the complex number j . (5.51) Lemma. Let 0 be an equilateral triangle contained in the interior of T with sides parallel to those of T . Then I I n H1 (z) dz = [Hτn (z)/τ ] dz + O(n − ) 0 I0 = [Hτn2 (z)/τ 2 ] dz + O(n − ). 0

Proof. Every triangular facet of Tn (that is, a triangular union of faces) points either upwards (in that its horizontal side is at its bottom) or downwards. Let 0 be an equilateral triangle contained in the interior of T with sides parallel to those of T , and assume that 0 points upwards (the same argument works for downwardpointing triangles). Let 0 n be the subgraph of Tn lying within 0, so that 0n is a triangular facet of Tn . Let D n be the set of downward-pointing faces of 0 n . Let η be a vector of R2 such that: if z is the centre of a face√of Dn then z + η is the centre of a neighbouring face, that is η ∈ {i, i τ, i τ 2 }/(n 3). Write h nj (z, η) = P[E jn (z + η) \ E jn (z)]. By Lemma 5.49, H1n (z + η) − H1n (z) = h n1 (z, η) − h n1 (z + η, −η)

= h nτ (z, ητ ) − h nτ (z + η, −ητ ).

c G. R. Grimmett 6 February 2009

[5.7]

Now,

Cardy formula

105

Hτn (z + ητ ) − Hτn (z) = h nτ (z, ητ ) − h nτ (z + ητ, −ητ ),

and so there is a cancellation in X X (5.52) Iηn = [H1n (z + η) − H1n (z)] − [Hτn (z + ητ ) − Hτn (z)] z∈D n

z∈D n

of all terms except those of the form h nτ (z 0 , −ητ ) for certain z 0 lying in faces of Tn that abut ∂0 n . There are O(n) such z 0 , and therefore, by Lemma 5.44, |Iηn | ≤ O(n 1− ).

(5.53) Consider the sum Jn =

1 n (I + τ Iinτ + τ 2 Iinτ 2 ), n i

where I jn is an abbreviation for I n

√ j/n 3

in (5.52). The terms of the form Hjn (z) in

(5.52) contribute 0 to J n , since each is multiplied by (1 + τ + τ 2 )n −1 = 0. The remaining terms of the form Hjn (z + η), Hjn (z + ητ ) mostly disappear also, and one is left only with terms Hjn (z 0 ) for certain z 0 at the centre of upwards-pointing faces of Tn abutting ∂0 n . For example, the contribution from z 0 if its face is at the bottom (but not the corner) of 0 n is  1 1 (τ + τ 2 )H1n (z 0 ) − (1 + τ )Hτn (z 0 ) = − [H1n (z 0 ) − Hτn (z 0 )/τ ]. n n

When z 0 is at the right (respectively, left) edge of 0 n , one obtains the same term multiplied by τ (respectively, τ 2 ). In summary, I (5.54) [H1n (z) − Hτn (z)/τ ] dz = −J n + O(n − ) = O(n − ), 0n

by (5.53), where the first O(n − ) term covers the fact that the z in (5.54) is a continuous rather than discrete variable. Since 0 and 0 n differ only around their boundaries, and the Hjn are uniformly H¨older, (5.55)

I

0

[H1n (z) − Hτn (z)/τ ] dz = O(n − )

and, by a similar argument, I (5.56) [H1n (z) − Hτn2 (z)/τ 2 ] dz = O(n − ). 0

The lemma is proved. c G. R. Grimmett 6 February 2009



106

[5.7]

Further Percolation

As remarked after the proof of Lemma (5.45), the sequence (H1n , Hτn , Hτn2 ) possesses subsequential limits, and it suffices for convergence to show that all such limits are equal. Let (H1 , Hτ , Hτ 2 ) be such a subsequential limit. By Lemma 5.51, the contour integrals of H1 , Hτ /τ , Hτ 2 /τ 2 around any 0 are equal. Therefore, the contour integrals of the G i in (5.48) around any 0 equal zero. By Morera’s theorem [2, 185], G 1 and G 2 are analytic on the interior of T , and furthermore they may be extended by continuity to the boundary of T . In particular, G 1 is analytic and real-valued, whence G 1 is a constant. By (5.46), G 1 (z) → 1 as z → 0, whence H1 + Hτ + Hτ 2 ≡ 1 on T . Therefore, the real part of G 2 satisfies

Re(G 2 ) = H1 − 21 (Hτ + Hτ 2 ) = 21 (3H1 − 1),

(5.57) and similarly (5.58)

2Re(G 2 /τ ) = 3Hτ − 1,

2Re(G 2 /τ 2 ) = 3Hτ 2 − 1.

Since the Hj are the real parts of analytic functions, they are harmonic. It remains to verify the relevant boundary conditions, and we will concentrate on the function Hτ 2 . There are two ways of doing this, of which the first specifies certain derivatives of the Hj along the boundary of T . By continuity, Hτ 2 (C) = 1 and Hτ 2 ≡ 0 on A B. We claim that the horizontal derivative, ∂ Hτ 2 /∂1, is 0 on AC ∪ BC. Once this is proved, it follows that Hτ 2 (z) is the unique harmonic√function on T satisfying these boundary conditions, namely the function 2|=(z)|/ 3. The remaining claim is proved as follows. Since G 2 is analytic, it satisfies the threefold Cauchy–Riemann equations (5.50). By (5.57)– (5.58),     2 ∂ H1 ∂ Hτ 2 2 1 ∂ G2 1 ∂ G2 = Re = (5.59) = Re . 2 3 ∂1 3 τ ∂1 3 τ ∂τ ∂τ Now, H1 ≡ 0 on BC, and BC has gradient τ , whence the right side of (5.59) 8 equals 0 on BC. The same argument holds on AC with H1 replaced by Hτ . The alternative is slightly simpler, see [25]. For z ∈ T , G 2 (z) is a convex combination of 1, τ , τ 2 , and thus maps T to the complex triangle T 0 with these three vertices. Furthermore, G 2 maps ∂ T to ∂ T 0 , and G 2 ( A j ) = j for j = 1, τ, τ 2 . Since G 2 is analytic on the interior of T , it is conformal, and there is a unique such conformal map with this boundary behaviour, namely that composed of a suitable dilation, rotation, and translation of T . This identifies G 2 uniquely, and the functions Hj also by (5.57)–(5.58). This concludes the proof of the Cardy formula when the domain D is an equilateral triangle. The proof for general D is essentially the same, on noting that 8 We

need also that G 2 may be continued analytically beyond the boundary of T , see [210].

c G. R. Grimmett 6 February 2009

[5.8]

The critical probability via the sharp-threshold theorem

107

a conformal image of a harmonic function is harmonic. First, one approximates to the boundary of D by a cycle of the triangular lattice with mesh δ. That G 1 (≡ 1) and G 2 are analytic is proved as before, and hence the corresponding limit functions H1 , Hτ , Hτ 2 are each harmonic with appropriate boundary conditions. We now apply conformal invariance. By the Riemann mapping theorem, there exists a conformal map φ from the inside of D to the inside of T that may be extended uniquely to their boundaries, and that maps a (respectively, b, c) to A (respectively, B, C). The triple (H1 ◦ φ −1 , Hτ ◦ φ −1 , Hτ 2 ◦ φ −1 ) solves the corresponding problem on T . We have seen that there is a unique such triple on T , given as above, and equation (5.43) is proved.

5.8 The critical probability via the sharp-threshold theorem We use the sharp-threshold Theorem 4.77 to prove the following. (5.60) Theorem [211]. The critical probability of site percolation on the triangular lattice satisfies pc = 12 . This may be proved in the same manner as Theorem 5.33, but we choose here to use the sharp-threshold theorem. This theorem provides a convenient ‘package’ for obtaining the steepness of a box-crossing probability, viewed as a function of p. Other means, more elementary and discovered earlier, may be used instead. These include: Kesten’s original proof [135] for bond percolation on the square lattice, Russo’s ‘approximate zero–one law’ [188], and, most recently, the proof of Smirnov presented in Section 5.6. Sharp-thresholds were first used in [42] in the current context, and later in [43, 89, 90]. The present proof may appear somewhat shorter than that of [42]. Proof. Let θ ( p) denote the percolation probability on the triangular lattice T. We have that θ ( 21 ) = 0, just as in the proof of the corresponding lower bound for the critical probability pc (L2 ) in Theorem 5.33, and we say no more about this. Therefore, pc ≥ 21 . Two steps remain. First, we shall use the sharp-threshold theorem to deduce that, when p > 12 , long rectangles are traversed with high probability in the long direction. Then we shall use that fact, within a block argument, to show that θ ( p) > 0. Each vertex is declared black (or open) with probability p, and white otherwise. In the notation introduced just prior to Lemma 5.28, let Hn = H16n,n √3 be the event that the rectangle Rn = R16n,n √3 is traversed by a black path in the long direction. By Lemmas 5.28–5.30, there exists τ > 0 such that (5.61)

φ 1 (Hn ) ≥ τ, 2

c G. R. Grimmett 6 February 2009

n ≥ 1.

108

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Further Percolation

x

Figure 5.20. The vertex x is pivotal for Hn if and only if: there is left–right black crossing of Rn when x is black, and a top–bottom white crossing when x is white.

Let x be a vertex of Rn , and write In, p (x) for the influence of x on the event Hn under the measure φ p , see (4.24). Now, x is pivotal for Hn if and only if: (i) the two vertical sides of Rn are connected by an open black path when x is black, (ii) the two horizontal sides of Rn are connected by an open white path when x is white. This event is illustrated in Figure 5.20. Let 21 ≤ p ≤ 43 , say. By (ii), (1 − p)In, p (e) ≤ φ1− p (rad(C 0 ) ≥ n), where

rad(C 0 ) = max{|x| : 0 ↔ x}

is the radius of the cluster at the origin. (Here, |x| denotes the graph-theoretic distance from x to the origin.) Since θ ( 21 ) = 0, φ1− p (rad(C 0 ) ≥ n) ≤ ηn where (5.62)

ηn = φ 1 (rad(C 0 ) ≥ n) → 0

as n → ∞.

2

We have used the fact that θ ( 21 ) = 0 here. By (5.61) and Theorem 4.77, for large n, φ 0p (Hn ) ≥ cτ (1 − φ p (Hn )) log[1/(8ηn )],

p ∈ [ 12 , 43 ],

which may be integrated to give (5.63)

1 − φ p (Hn ) ≤ (1 − τ )[8ηn ]cτ ( p− 2 ) ,

c G. R. Grimmett 6 February 2009

1

p ∈ [ 12 , 34 ].

[5.8]

The critical probability via the sharp-threshold theorem

109

Figure 5.21. A block is declared ‘red’ if it contains open paths that: √ (i) traverse it in the long direction, and (ii) traverse it in the short direction within the 3n × n 3 region at each end of the block. The shorter crossings exist if the inclined blocks are traversed in the long direction.

Let p > 21 . By (5.62)–(5.63), (5.64)

φ p (Hn ) → 1

as n → ∞.

We turn to the required block argument, which differs from that of Section 5.6 in that we shall use no explicit estimate of φ p (Hn ). Roughly speaking, copies of the rectangle Rn are distributed about T in such a way that each copy corresponds to an edge of a re-scaled copy of T. The detailed construction of this ‘renormalized block lattice’ is omitted from these notes, and we rely on Figure 5.22 for explanation. The ‘blocks’ (that is, the copies of Rn ) are in one–one correspondence with the edges of T, and thus we may label the blocks as Be , e ∈ ET . Each block intersects just ten other blocks. Next we define a ‘block event’, that is, a certain event defined on the configuration within a block. The first requirement for this event is that the block be traversed by an open path in the long direction. We shall require some further paths in order that, when two such blocks intersect, their union contains a single component that traverses each in its long direction. In specific, we require open paths traversing the block in the short direction, within each of the two extremal √ 3n × n 3 regions of the block. A block is coloured red if the above paths exist within it. See Figure 5.21. If two red blocks, Be and B f say, are such that e and f share a vertex, then their union possesses a single open component containing paths traversing each of Be and B f . If the block Rn fails to be red, then one or more of the blocks in Figure 5.21 is not traversed by an open path in the long direction. Therefore, ρ n := φ p (Rn is red) c G. R. Grimmett 6 February 2009

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Figure 5.22. Each block is red with probability ρn . There is an infinite cluster of red blocks with strictly positive probability, and any such cluster contains an infinite open cluster of the original lattice.

satisfies (5.65)

1 − ρn ≤ 3[1 − φ p (Hn )] → 0

as n → ∞,

by (5.64). The states of different blocks are dependent random variables, but any collection of disjoint blocks have independent states. We shall count paths in the dual, as in (3.8), to obtain that there exists, with strictly positive probability, an infinite path in T comprising edges e such that every such Be is red. This implies the existence of a infinite open cluster in the original lattice. If the red cluster at the origin of the block lattice is finite, there exists a path in the dual lattice (a copy of the hexagonal lattice) that crosses only non-red blocks (as in Figure 3.2). Within any dual path of length m, there exists a set of bm/12c or more edges such that the corresponding blocks are pairwise disjoint. Therefore, the probability that the origin of the block lattice lies in a finite cluster only of red blocks is no greater than ∞ X

m=6

3m (1 − ρn )bm/12c .

By (5.65), this may be made smaller than 21 by choosing n sufficiently large. Therefore, θ ( p) > 0 for p > 21 , and the theorem is proved. 

c G. R. Grimmett 6 February 2009

[5.9]

Exercises

111

5.9 Exercises 5.1. [32] Consider bond percolation on Z2 with p = 21 , and define the radius  of the open cluster C at the origin by rad(C) = max n : 0 ↔ ∂[−n, n]2 . Use the BK inequality to show that 1




P 1 rad(C) ≥ n ≥ √ . 2

2 n

5.2. Let Dn be the largest diameter (in the sense of graph theory) of the open clusters of bond percolation on Zd that intersect the box [−n, n]d . Show when p < pc that Dn / log n → α( p) almost surely and in L p , for some α( p) ∈ (0, ∞). 5.3. Consider bond percolation on L2 with density p. Let Tn be the box [0, n]2 with periodic boundary conditions, that is, we identify any pair (u, v), (x, y) satisfying: either u = 0, x = n, v = y, or v = 0, y = n, u = x. For given m < n, let A be the event that some translate of [0, m]2 in Tn is crossed by an open path either from top to bottom, or from left to right. Using the theory of influence or otherwise, show that P p ( A) ≥ 1 − 21 m −c( p− 2 ) , 1

p > 12 .

5.4. Consider site percolation on the triangular lattice T, and let B(n) be the ball of radius n centred at the origin. Use the RSW theorem to show that P 1 (0 ↔ ∂ B(n)) ≥ cn −α 2

for constants c, α > 0. Using the coupling of Section 3.3 or otherwise, deduce that θ ( p) ≤ c 0 ( p − 21 )β for p > 12 and constants c0 , β > 0. 5.5. By adapting the arguments of Section 5.5 or otherwise, develop an RSW theory for bond percolation on Z2 . 5.6. Let D be an open simply connected domain in R2 whose boundary ∂ D is a Jordan curve. Let a, b, x, c be distinct points on ∂ D taken in anticlockwise order. Let Pδ (ac ↔ bx) be the probability that, in site percolation on the re-scaled triangular lattice δ T with density 21 , there exists an open path within D ∪ ∂ D from some point on the arc ac to some point on bx. Show that Pδ (ac ↔ bx) is uniformly bounded away from 0 and 1 as δ → 0. 5.7. Let f : D → C where D is an open simply-connected region of the complex plane. If f is C 1 and satisfies the threefold Cauchy–Riemann equations (5.50), show that f is analytic.

c G. R. Grimmett 6 February 2009

6 Contact Model

The contact process is a model for the spread of disease about the vertices of a graph. It has a property of duality that arises through the reversal of time. For a vertex-transitive graph such as the d-dimensional lattice, there is a multiplicity of invariant measures if and only if there is a strictly positive probability of an unbounded path of infection in space–time starting from a given vertex. This observation permits the use of methodology developed for the study of oriented percolation. When the underlying graph is a tree, the model has three distinct phases, termed extinction, weak survival, and strong survival. There is a continuous-time percolation model that differs from the contact model in that the time axis is undirected.

6.1 Stochastic epidemics One of the simplest stochastic models for the spread of an epidemic is as follows. Consider a population of constant size N + 1 that is suffering from an infectious disease. We can model the spread of the disease as a Markov process. Let X (t) be the number of healthy individuals at time t and suppose that X (0) = N . We assume that, if X (t) = n, then the probability of a new infection during (t, t + h) is proportional to the number of possible encounters between ill folk and healthy folk. That is,
P X (t + h) = n − 1 X (t) = n = λn(N + 1 − n)h + o(h) as h ↓ 0. In the simplest situation, we assume that nobody recovers. It is easy to show that G(s, t) = E(s satisfies

X (t)

)=

N X n=0

s n P(X (t) = n)

  ∂G ∂2G ∂G = λ(1 − s) N −s 2 ∂t ∂s ∂s

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with G(s, 0) = s N . There is no simple way of solving this equation, though a lot of information is available about approximate solutions. This epidemic model is over-simplistic through the assumptions that: – the process is Markovian, – there are only two states and no recovery, – there is total mixing, in that the rate of spread is proportional to the product of the numbers of infectives and susceptibles. In ‘practice’ (computer viruses apart), one infects only individuals in one’s immediate (bounded) vicinity. The introduction of spatial relationships into such a model adds a major complication, and is achieved through so-called ‘contact model’ of Harris [122]. Let G = (V, E) be a (finite or infinite) graph with bounded vertex-degrees. The contact model on G is a continuous-time Markov process on the state space 6 = {0, 1}V . A state is therefore a 0/1 vector ξ = (ξ(x) : x ∈ V ), where 0 represents the healthy state and 1 the ill state. There are two parameters: an infection rate λ and a recovery rate δ. Transition-rates are given informally as follows. Suppose that the state at time t is ξ ∈ 6, and let x ∈ V . Then P(ξt+h (x) = 0 | ξt = ξ ) = δh + o(h),

P(ξt+h (x) = 1 | ξt = ξ ) = λNξ (x)h + o(h),

if ξ(x) = 1,

if ξ(x) = 0,

where Nξ (x) is the number of neighbours of x that are infected in ξ : Nξ (x) = {y ∈ V : y ∼ x, ξ(y) = 1} .

Thus, each ill vertex recovers at rate δ, and in the meantime infects any given neighbour at rate λ. Care is needed when specifying a Markov process through its transition rates, especially when G is infinite, since then 6 is uncountable. We shall see in the next section that the contact model can be constructed via a countably infinite collection of Poisson processes. More general approaches to the construction of interacting particle processes are described in [148] and summarized in Section 10.1.

6.2 Coupling and duality The contact model can be constructed in terms of families of Poisson processes. This representation is both informative and useful for what follows. For each x ∈ V we draw a ‘time-line’ [0, ∞). On the time-line {x} × [0, ∞) we place a Poisson point process D x with intensity δ. For each ordered pair x, y ∈ V of neighbours, we let B x,y be a Poisson point process with intensity λ. These processes are taken to be independent of each other, and we can assume without c G. R. Grimmett 6 February 2009

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time space 0 Figure 6.1. The so called ‘graphical representation’ of the contact process on the line . The horizontal line represents ‘space’, and the vertical line above a point x is the time-line at x. The marks ◦ are the points of cure, and the arrows are the arrows of infection. Suppose we are told that, at time 0, the origin is the unique infected point. In this picture, the initial infective is marked 0, and the bold lines indicate the portions of space–time which are infected.

loss of generality that the times occurring in the processes are distinct. Points in each Dx are called ‘points of cure’, and points in B x,y are called ‘arrows of infection’ from x to y. The appropriate probability measure is denoted by P λ,δ . The situation is illustrated in Figure 6.1 with G = L. Let (x, s), (y, t) ∈ V × [0, ∞) where s ≤ t. We define a (directed) path from (x, s) to (y, t) to be a sequence (x, s) = (x 0 , t0 ), (x 0 , t1 ), (x 1 , t1 ), (x 1 , t2 ), . . . , (x n , tn+1 ) = (y, t) with t0 ≤ t1 ≤ · · · ≤ tn+1 , such that: 1. each interval {xi } × [ti , ti +1 ] contains no points of D xi , 2. ti ∈ Bxi−1 ,xi for i = 1, 2, . . . , n. We write (x, s) → (y, t) if there exists such a directed path. We think of a point (x, u) of cure as meaning that an infection at x just prior to time u is cured at time u. A arrow of infection from x to y at time u means that an infection at x just prior to u is passed at time u to y. Thus, (x, s) → (y, t) means that y is infected at time t if x is infected at time s. Let ξ0 ∈ 6 = {0, 1}V , and define ξt ∈ 6, t ∈ [0, ∞), by ξt (y) = 1 if and only if there exists x ∈ V such that ξ0 (x) = 1 and (x, 0) → (y, t). It is clear that (ξt : t ∈ [0, ∞)) is a contact model with parameters λ and δ. The above ‘graphical representation’ has several uses. First, it is a geometrical picture of the spread of infection that provides a coupling of contact models with all possible initial configurations ξ0 . Secondly, it provides couplings of contact models with different λ and δ, as follows. Let λ1 ≤ λ2 and δ1 ≥ δ2 , and consider the above representation with (λ, δ) = (λ2 , δ1 ). If we remove each point of cure with probability δ2 /δ1 (respectively, each arrow of infection with probability c G. R. Grimmett 6 February 2009

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115

λ1 /λ2 ), we obtain a representation of a contact model with parameters (λ 2 , δ2 ) (respectively, parameters (λ1 , δ1 )). We obtain thus that the passage of infection is non-increasing in δ and non-decreasing in λ. There is a natural one–one correspondence between 6 and the power set 2 V of the vertex-set, given by ξ ↔ Iξ = {x ∈ V : ξ(x) = 1}. We shall frequently regard vectors ξ as sets Iξ . For ξ ∈ 6 and A ⊆ V , we write ξtA for the value of the contact model at time t starting at time 0 from the set A of infectives. It is immediate by the rules of the above coupling that: (a) the coupling is monotone in that ξtA ⊆ ξtB if A ⊆ B, (b) the coupling is additive in that ξtA∪B = ξtA ∪ ξtB .

(6.1) Theorem. Duality relation. For A, B ⊆ V , (6.2)

Pλ,δ (ξtA ∩ B 6= ∅) = Pλ,δ (ξtB ∩ A 6= ∅).

Equation (6.2) can be written in the form B A (ξt ≡ 0 on A). (ξt ≡ 0 on B) = Pλ,δ Pλ,δ

Proof. This hinges on the fact that a Poisson process remains a Poisson process when time is run backwards. The event on the left side of (6.2) is the union over a ∈ A and b ∈ B of the event that (a, 0) → (b, t). If we reverse the direction of time, and the directions of the arrows of infection, the probability of this event is unchanged and it corresponds now to the event on the right side of (6.2). 

6.3 Invariant measures and percolation In this and the next section, we consider the contact model ξ = (ξ t : t ≥ 0) when the underlying graph is the d-dimensional cubic lattice Ld , with d ≥ 1. d Thus, ξ is a Markov process on the state space 6 = {0, 1}Z . Let I be the set of invariant measures of ξ , that is, the set of probability measures µ on 6 such that µSt = µ, where S = (St : t ≥ 0) is the transition semigroup of the process. It is elementary that I is a convex set of measures: if φ1 , φ2 ∈ I, then αφ1 + (1 − α)φ2 ∈ I for α ∈ [0, 1]. Therefore, I is determined by knowledge of the set Ie of extremal invariant measures. A further discussion of the transition semigroup and its relationship to invariant measures can be found in Section 10.1. The partial order on 6 induces a partial order on probability measures on 6 in the usual way, and we denote this by ≤st . It turns out that I possesses a ‘minimal’ and ‘maximal’ element, with respect to ≤st . The minimal measure (or ‘lower invariant measure’) is the measure that places probability 1 on the empty set, denoted δ∅ . It is called ‘lower’ because δ∅ ≤st µ for all measures µ on 6. c G. R. Grimmett 6 February 2009

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The maximal measure (or ‘upper invariant measure’) is constructed as the weak limit of the contact model beginning with the set ξ0 = Zd . Let µs denote the law d d of ξsZ . Since ξsZ ⊆ Zd , µ0 Ss = µs ≤st µ0 .

By the monotonicity of the coupling,

µs+t = µ0 Ss St = µs St ≤st µt , whence the limit

lim µt ( f )

t→∞

exists for any bounded increasing function f : 6 → R. Using the compactness of (6, F ) and a result from measure theory, the weak limit ν = lim µt t→∞

exists, and is called the upper invariant measure. It is clear by the method of its construction that ν is invariant under the action of any translation of L d . (6.3) Proposition. We have that δ∅ ≤st ν ≤st ν for every ν ∈ I.

Proof. Let ν ∈ I. The first inequality is trivial. Clearly, ν ≤st µ0 , since µ0 is concentrated on the maximal set Zd . By the monotonicity of the coupling, ν = ν St ≤st µ0 St = µt ,

t ≥ 0.

Let t → ∞ to obtain that ν ≤st ν.



By Proposition 6.3, there exists a unique invariant measure if and only if ν = δ ∅ . In order to understand when this is so, we deviate briefly to consider a percolationtype question. Suppose we begin the process at a singleton, the origin say, and ask whether the probability of survival for all time is strictly positive. That is, we work with the percolation-type probability θ (λ, δ) = Pλ,δ (ξt0 6= ∅ for all t ≥ 0),

(6.4)

where ξt0 = ξt . By a re-scaling of time, θ (λ, δ) = θ (λ/δ, 1), and we assume henceforth in his section that δ = 1, and we write Pλ = Pλ,1 . {0}

(6.5) Proposition. The density of ill vertices under ν equals θ (λ). That is,
θ (λ) = ν {σ ∈ 6 : σx = 1} , x ∈ Zd . Proof. The event that ξ T0 ∩ Zd 6= ∅ is non-increasing in T , whence θ (λ) = lim Pλ (ξT0 ∩ Zd 6= ∅). T →∞

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117

By Proposition 6.1, Pλ (ξT0 ∩ Zd 6= ∅) = Pλ (ξTZ (0) = 1), d

and by weak convergence,


Pλ (ξTZ (0) = 1) → ν {σ ∈ 6 : σ0 = 1} . d

The claim follows by the translation-invariance of ν.



We define the critical value of the process by λc = λc (d) = sup{λ : θ (λ) = 0}. The function θ (λ) is non-decreasing, so that θ (λ) By Proposition 6.5, ν





=0

>0

= δ∅

6= δ∅

if λ < λc , if λ > λc . if λ < λc , if λ > λc .

The case λ = λc is delicate, especially when d ≥ 2, and it has been shown in [33], using a slab argument related to that of the proof of Theorem 5.17, that θ (λ c ) = 0 for d ≥ 1. (6.6) Theorem [33]. Consider the contact model on Ld with d ≥ 1. The set I of invariant measures comprises a singleton if and only if λ ≤ λc . That is, I = {δ∅ } if and only if λ ≤ λc . There are further consequences of the arguments of [33] of which we mention one. The geometrical constructions of [33] enable a proof of the equivalent for the contact model of the ‘slab’ percolation Theorem 5.17. This in turn completes the proof, initiated in [69, 73], that the set of extremal invariant measures of the contact model on Ld is exactly Ie = {δ∅ , ν}. See [71] also.

6.4 The critical value This section is devoted to the following theorem 1 . Recall that the rate of cure is taken as δ = 1. 1 There

are physical reasons to suppose that λc (1) = 1.6494 . . . , see the discussion of the so-called reggeon spin model in [91, 148]. c G. R. Grimmett 6 February 2009

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(6.7) Theorem [122]. For d ≥ 1, we have that (2d)−1 < λc (d) < ∞.

The lower bound is easily improved to λc (d) ≥ (2d − 1)−1 . The upper bound may be refined to λc (d) ≤ d −1 λc (1) < 1, as indicated in Exercise 6.2. See the accounts of the contact model in the two Liggett volumes [148, 150]. Proof. The lower bound is obtained by a random walk argument. The integervalued process Nt = |ξt0 | decreases by 1 at rate N t . It increases by 1 at rate λTt where Tt is the number of edges of Ld exactly one of whose endvertices x satisfies ξt0 (x) = 1. Now, Tt ≤ 2d Nt , and so the jump-chain of N t is bounded above by a simple random walk R = (Rn : n ≥ 0) on {0, 1, 2, . . . }, with absorption at 0, and that moves to the right with probability p= at each step. It is elementary that

2dλ 1 + 2dλ

P(Rn = 0 for some n ≥ 0) = 1

and it follows that θ (λ) = 0

if λ <

if

p ≤ 12 ,

1 . 2d

Just as in the case of percolation (Theorem 3.2) the upper bound on λ c requires more work. Since Zd may be embedded in Z, it is elementary that λc (d) ≤ λc (1). We show by a discretization argument that λc (1) < ∞. Let 1 > 0, and let m, n ∈ Z be such that m + n is even. We shall define independent random variables X m,n taking the values 0 and 1. We declare X m,n = 1, and call (m, n) open, if and only if, in the graphical representation of the contact model, the following two events occur:  (a) there is no point of cure in the interval {m} × (n − 1)1, (n + 1)1 , (b) there exist left and  right pointing arrows of infection from the interval {m} × n1, (n + 1)1 . It is immediate that the X m,n are independent, and p = p(1) = Pλ (X m,n = 1) = e−21 (1 − e−λ1 )2 . We choose 1 to maximize p(1), which is to say that e−λ1 = and (6.8) c G. R. Grimmett 6 February 2009

p=

1 , 1+λ

λ2 . (1 + λ)2+2/λ

[6.5]

The contact model on a tree

119

0

Figure 6.2. Part of the binary tree T2 .

Consider the X m,n as giving rise to a directed site percolation model on the first 0 quadrant of a rotated copy of Z2 . It can be seen that ξn1 ⊇ Bn , where Bn is the set of vertices of the form (m, n) that are reached from (0, 0) along open paths of the percolation process. Now,
Pλ |Bn | = ∞ for all n ≥ 0 > 0 if p > pEcsite where pEcsite is the critical probability of the percolation model. By (6.8), θ (λ) > 0

if

λ2 > pEcsite . 2+2/λ (1 + λ)

Since2 pEcsite < 1, the final inequality is valid for sufficiently large λ, and we have proved that λc (1) < ∞. 

6.5 The contact model on a tree Let d ≥ 2 and let Td be the homogeneous (infinite) labelled tree in which every vertex has degree d + 1, illustrated in Figure 6.2. We identify a distinguished vertex, called the origin and denoted 0. Let ξ = (ξt : t ≥ 0) be a contact model on Td with infection rate λ and initial state ξ0 = {0}, and take δ = 1. With θ (λ) = Pλ (ξt 6= ∅ for all t), the process is said to die out if θ (λ) = 0, and to survive if θ (λ) > 0. It is said to survive strongly if
Pλ ξt (0) = 1 for unbounded times t > 0, 2 Exercise.

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and to survive weakly if it survives but it does not survive strongly. A process that survives weakly has the property that (with strictly positive probability) the illness exists for all time, but that (almost surely) there is a final time at which any given vertex is infected. It can be shown that weak survival never occurs on a lattice L d , see [150]. The picture is quite different on a tree. The properties of survival and strong survival are evidently non-decreasing in λ, whence there exist values λc , λss satisfying λc ≤ λss such that the process dies out if survives weakly if survives strongly if

λ < λc , λc < λ < λss , λ > λss .

When is it the case that λc < λss ? The next theorems indicate that this occurs on Td if d ≥ 6. It was further proved in [171] that strict inequality holds whenever d ≥ 3, and this was extended in [149] to d ≥ 2. See [150, Chap. I.4] and the references therein. (6.9) Theorem [171]. For the contact model on the tree Td with d ≥ 2, λc <

1 . d −1

(6.10) Theorem [171]. For the contact model on the tree Td with d ≥ 2, 1 λss ≥ √ . 2 d Proof of Theorem 6.9. Let ρ ∈ (0, 1), and νρ ( A) = ρ |A| for any finite subset A of the vertex-set V of Td . We shall observe the process νρ (ξt ). Let g A (t) = EλA (νρ (ξt )). It is an easy calculation that (6.11)

as t ↓ 0, where

   νρ ( A) + λN t ρνρ ( A) g (t) = | A|t ρ + (1 − | A|t − λN t)νρ ( A) + o(t), A



N = {hx, yi : x ∈ A, y ∈ / A}

is the number of edges of Td with exactly one endvertex in A. Now, (6.12)

N ≥ (d + 1)| A| − 2(| A| − 1),

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The contact model on a tree

121

since there are no more than | A| − 1 edges having both endvertices in A. By (6.11),   | A| d A = (1 − ρ) (6.13) g (t) − λN νρ ( A) dt ρ t=0  
| A| ≤ (1 − ρ)νρ ( A) 1 − λρ(d − 1) − 2λ ρ ≤ −2λ(1 − ρ)νρ ( A) ≤ 0, whenever (6.14)

λρ(d − 1) ≥ 1.

Assume that (6.14) holds. By (6.13) and the Markov property,   d ξu d A A ≤ 0, (6.15) g (u) = Eλ g (t) du dt t=0

implying that g A (u) is non-increasing in u. With A = {0}, we have that g(0) = ρ < 1, and therefore lim t→∞ g(t) ≤ ρ. On the other hand, if the process dies out, then (almost surely) ξ t = ∅ for all large t, so that, by the bounded convergence theorm, g(t) → 1 as t → ∞. From this contradiction, we deduce that the process survives whenever there exists ρ ∈ (0, 1) such that (6.14) holds. The theorem is proved.  Proof of Theorem 6.10. Once again, take ρ ∈ (0, 1). We draw the tree in the manner of Figure 6.3, and we let l(x) be the generation number of the vertex x in this representation. For a finite subset A of V , let X wρ ( A) = ρ l(x) , x∈A

with the convention that an empty summation equals 0. As in (6.13), h A (t) = EλA (wρ (ξt )) satisfies X X d A (6.16) = −ρ l(x) + λ h (t) dt t=0

y∈V : y∼x, y∈ /A

x∈A

≤ −wρ ( A) + λ = (λdρ + λρ

−1

X

x∈A

1 ρ=√ , d

c G. R. Grimmett 6 February 2009



ρ l(x) [dρ + ρ −1 ]

− 1)wρ ( A).

Set (6.17)

ρ

l(y)

1 λ= √ , 2 d

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Contact Model

l(x) = −1 l(x) = 0

0

l(x) = 1 l(x) = 2

Figure 6.3. The binary tree T2 ‘suspended’ from a given doubly-infinite path, with the generation numbers as marked.

so that λdρ + λρ −1 − 1 = 0. By (6.16), wρ (ξt ) is a positive supermartingale. By the martingale convergence theorem, the limit (6.18)

M = lim wρ (ξt ), t→∞

exists PλA -almost surely. See Section 12.3 of [109] for an account of the convergence of martingales. On the event I = {ξt (0) = 1 for unbounded times t}, the process wρ (ξt ) changes its value (almost surely) by ρ 0 = 1 on an unbounded set of times t, in contradiction of (6.18). Therefore, PλA (I ) = 0, and the process does not converge strongly under (6.17). The theorem is proved. 

6.6 Space–time percolation The percolation models of Chapters 2 and 5 are discrete in that they inhabit a discrete graph G = (V, E). There are a variety of continuum models of interest (see [99] for a summary) of which we distinguish the continuum model on V × R. One can consider this as the contact model with undirected time. We will encounter the related continuum random-cluster model in Chapter 9, together with its application to the quantum Ising model. Let G = (V, E) be a finite graph. The percolation model of this section inhabits the space V × R, which we refer to as space–time, and we think of V × R as being obtained by attaching a ‘time-line’ (−∞, ∞) to each vertex x ∈ V . c G. R. Grimmett 6 February 2009

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123

Let λ, δ ∈ (0, ∞). The continuum percolation model on V × R is constructed via processes of ‘cuts’ and ‘bridges’ as follows. For each x ∈ V , we select a Poisson process Dx of points in {x} × R with intensity δ; the processes {D x : x ∈ V } are independent, and the points in the D x are termed ‘cuts’. For each e = hx, yi ∈ E, we select a Poisson process Be of points in {e} × R with intensity λ; the processes {Be : e ∈ E} are independent of each other and of the D x . Let φλ,δ denote the probability measure associated with the family of such Poisson processes indexed by V ∪ E. For each e = hx, yi ∈ E and (e, t) ∈ Be , we think of (e, t) as an edge joining the endpoints (x, t) and (y, t), and we refer to this edge as a ‘bridge’. For (x, s), (y, t) ∈ V × R, we write (x, s) ↔ (y, t) if there exists a path π with endpoints (x, s), (y, t) such that: π is a union of cut-free sub-intervals of V × R and bridges. For 3, 1 ⊆ V × R, we write 3 ↔ 1 if there exist a ∈ 3 and b ∈ 1 such that a ↔ b. For (x, s) ∈ V × R, let C x,s be the set of all points (y, t) such that (x, s) ↔ (y, t). The clusters C x,s have been studied in [34], where the case G = Zd was considered in some detail. Let 0 denote the origin (0, 0) ∈ Zd × R, and let C = C 0 denote the cluster at the origin. Noting that C is a union of line-segments, we write |C| for its Lebesgue measure. The radius rad(C) of C is given by

where

 rad(C) = sup kxk + |t| : (x, t) ∈ C ,

kxk = sup |xi |, i

x = (x 1 , x 2 , . . . , x d ) ∈ Zd ,

is the supremum norm on Zd . The critical point of the process is defined by λc (δ) = sup{λ : θ (λ, δ) = 0}, where θ (λ, δ) = Pλ,δ (|C| = ∞).

It is immediate by re-scaling time that θ (λ, δ) = θ (λ/δ, 1), and we shall use the abbreviations λc = λc (1) and θ (λ) = θ (λ, 1). (6.19) Theorem [34]. Let G = Ld where d ≥ 1, and consider continuum percolation on Zd × R. (a) Let λ, δ ∈ (0, ∞). There exist γ , ν satisfying γ , ν > 0 for λ/δ < λc such that: Pλ,δ (|C| ≥ k) ≤ e−γ k ,

Pλ,δ (rad(C) ≥ k) ≤ e−νk , c G. R. Grimmett 6 February 2009

k > 0, k > 0.

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[6.6]

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(b) When d = 1, λc = 1 and θ (1) = 0.

There is a natural duality in 1 + 1 dimensions (that is, when the underlying graph is the line L), and it is easily seen in this case that the process is self-dual when λ = δ. Part (b) identifies this self-dual point as the critical point. For general d ≥ 1, the continuum percolation model on Ld × R has exponential decay of connectivity when λ/δ < λc . The theorem is proved by an adaptation to the continuum of the methods used for Ld+1 . Theorem 6.19 will be useful for the study of the quantum Ising model in Section 9.4. There has been considerable interest in the behaviour of the continuum percolation model on a graph G when the environment is itself chosen at random, that is, we take the λ = λe , δ = δx to be random variables. More precisely, suppose that the Poisson process of cuts at a vertex x ∈ V has some intensity δx , and that of bridges parallel to the edge e = hx, yi ∈ E has some intensity λe . Suppose further that the δ x , x ∈ V , are independent, identically distributed random variables, and the λe , e ∈ E also. Write 1 and 3 for independent random variables having the respective distributions, and P for the probability measure governing the environment. [As before, Pλ,δ denotes the measure associated with the percolation model in the given environment. The above use of the letters 1, 3 to denote random variables is temporary only.] The problem of understanding the behaviour of the system is now much harder, because of the fluctuations in intensities about G. If there exist λ0 , δ 0 ∈ (0, ∞) such that λ0 /δ 0 < λc and P(3 ≤ λ0 ) = P(1 ≥ δ 0 ) = 1, then the process is almost surely dominated by the subcritical percolation process with parameters λ0 , δ 0 , whence there is (almost sure) exponential decay in the sense of Theorem 6.19(i). This can fail in an interesting way if there is no such almostsure domination, in that (under certain conditions) one can prove exponential decay in the space-direction but only a weaker decay in the time-direction. The problem arises since there will generally be regions of space that are favourable to the existence of large clusters, and other regions that are unfavourable. In a favourable region, there may be unnaturally long connections between two points with similar values for their time-coordinates. For (x, s), (y, t) ∈ Zd × R and q ≥ 1, we define  dq (x, s; y, t) = max kx − yk, [log(1 + |s − t|)]q . (6.20) Theorem [137, 138]. Let G = Ld where d ≥ 1. Suppose that n
o
K = max P [log(1 + 3)]β , P [log(1 + 1−1 )]β < ∞,

√
for some β > 2d 2 1 + 1 + d −1 + (2d)−1 . There exists Q = Q(d, β) > 1 such that the following holds. For q ∈ [1, Q) and m > 0, there exists  = c G. R. Grimmett 6 February 2009

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Exercises

125

(d, β, K , m, q) > 0 and η = η(d, β, q) > 0 such that: if P



log(1 + (3/1))

β 

< ,

there exist identically distributed random variables D x ∈ L η (P), x ∈ Zd , such that




Pλ,δ (x, s) ↔ (y, t) ≤ exp −mdq (x, s; y, t)

for (x, s), (y, t) ∈ Zd × R.



if dq (x, s; y, t) ≥ Dx ,

This version of the theorem of Klein can be found with explanation in [106]. It is proved by a so-called multiscale analysis. Mention related results for contact models etc.

6.7 Exercises 6.1. Show that the critical probability of oriented site percolation on L 2 satisfies < 1. 6.2. Let d ≥ 2, and 5 : Zd → Z be given by

pEcsite

5(x 1 , x 2 , . . . , x d ) =

d X

xi .

i =1

Let At denote a contact model on Zd with parameter λ and starting at the origin. Show that one can couple A with a contact model C on Z, with parameter λd and starting at the origin, in such a way that 5( A t ) ⊇ C t for all t. Deduce that the critical point λc (d) of the contact model on Ld satisfies λc (d) ≤ −1 d λc (1). 6.3. [34] By adapting the corresponding argument for bond percolation on L2 , or otherwise, show that the percolation probability of unoriented space-time percolation on Z × R satisfies θ (λ, λ) = 0 for λ > 0.

c G. R. Grimmett 6 February 2009

7 Gibbs States

Brook’s theorem states that a positive probability measure on a finite product may be decomposed into factors indexed by the cliques of its dependency graph. Closely related to this is the well known fact that a positive measure is a spatial Markov field if and only if it is a Gibbs state. The Ising and Potts models are introduced, and the n-vector model is mentioned.

7.1 Dependency graphs Let X = (X 1 , X 2 , . . . , X n ) be a family of random variables on a given probability space. For i, j ∈ V = {1, 2, . . . , n} with i 6= j , we write i ⊥ j if: X i and X j are independent conditional on (X k : k 6= i, j ). The relation ⊥ is thus symmetric, and it gives rise to a graph G with vertex set V and edge-set E = {hi, j i : i 6⊥ j }, called the dependency graph of X (or of its law). We shall see that the law of X may be expressed as a product over terms corresponding to complete subgraphs of G. A complete subgraph of G is called a clique, and we write K for the set of all cliques of G. For notational simplicity later, we designate the empty subset of V to be a clique, and thus ∅ ∈ K. A clique is maximal if no strict superset is a clique, and we write M for the set of maximal cliques of G. We assume for simplicity that the X i take values in some countable subset S of the reals R. The law of X gives rise to a probability mass function π on S n given by π(x) = P(X i = xi for i ∈ V ), x = (x 1 , x 2 , . . . , x n ) ∈ S n . It is easily seen by the definition of independence that i ⊥ j if and only if π may be factorized in the form π(x) = f (xi , U )g(x j , U ),

x ∈ Sn ,

for some functions f and g, where U = (x k : k 6= i, j ). For K ∈ K and x ∈ S n , we write x K = (xi : i ∈ K ). We call π positive if π(x) > 0 for all x ∈ S n . In the following, each function f K acts on S K . c G. R. Grimmett 6 February 2009

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Dependency graphs

127

(7.1) Theorem [49]. Let π be a positive probability mass function on S n . There exist functions f K : S K → [0, ∞), K ∈ M, such that Y (7.2) π(x) = f K (x K ), x ∈ Sn . K ∈M

In the simplest non-trivial example, let us assume that i ⊥ j whenever |i − j | ≥ 2. The maximal cliques are the pairs hi, i + 1i, and the mass function π may be expressed in the form π(x) =

n−1 Y i =1

fi (xi , xi +1 ),

x ∈ Sn ,

so that X is a Markov chain, whatever the direction of time. Proof. We shall show that π may be expressed in the form Y (7.3) π(x) = f K (x K ), x ∈ Sn , K ∈K

for suitable f K . Representation (7.2) follows from (7.3) by associating each f K with some maximal clique K 0 that contains the clique K as a subset. A representation of π in the form Y π(x) = fr (x) r

is said to separate i and j if every fr is a constant function of either x i or x j , that is, no fr depends non-trivially on both x i and x j . Let Y (7.4) π(x) = f A (x A ) A∈A

be a factorization of π for some family A of subsets of V , and suppose that i , j satisfies: i ⊥ j , but i and j are not separated in (7.4). We shall construct from (7.4) a factorization that separates every pair r , s that is separated in (7.4), and in addition separates i , j . Continuing by iteration, we obtain a factorization that separates every pair i , j satisfying i ⊥ j , and this has the required form (7.3). Since i ⊥ j , π may be expressed in the form (7.5)

π(x) = f (xi , U )g(x j , U )

for some f , g, where U = (x k : j = 6 i, j ). Fix s ∈ S, and write h s for the function h(x) evaluated with x j = s. By (7.4), Y  π(x) π(x) , = f A (x A ) s (7.6) π(x) = π(x) s π(x) π(x) s

A∈A

s

and, by (7.5),

g(x j , U ) π(x) = g(s, U ) π(x) s

is independent of xi . Equation (7.6) is thus the required representation, and the claim is proved.  c G. R. Grimmett 6 February 2009

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[7.2]

Gibbs States

7.2 Markov fields and Gibbs states Let G = (V, E) be a finite graph, taken for simplicity without loops or multiple edges. Within statistics and statistical mechanics, there has been a great deal of interest in probability measures having a type of ‘spatial Markov property’ given in terms of the neighbour relation of G. We shall restrict ourselves here to measures on the sample space 6 = {0, 1} V , while noting that the following results may extended without material difficulty to a larger product S V where S is finite or countably infinite. The vector σ ∈ 6 may be placed in one–one correspondence with the subset η(σ ) = {v ∈ V : σv = 1} of V , and we shall use this correspondence freely. For any W ⊆ V , we define the external boundary 1W = {v ∈ V : v ∈ / W, v ∼ w for some w ∈ W }. For s = (sv : v ∈ V ) ∈ 6, we write sW for the sub-vector (sw : w ∈ W ). We refer to the configuration of vertices in W as the ‘state’ of W . (7.7) Definition. A probability measure π on 6 is said to be positive if π(σ ) > 0 for all σ ∈ 6. It is called a Markov field if it is positive and: for all W ⊆ V , conditional on the state of V \ W , the law of the state of W depends only on the state of 1W . That is, π satisfies the global Markov property:

(7.8) π σW = sW σV \W = sV \W = π σW = sW σ1W = s1W ,

for all s ∈ 6, and W ⊆ V .

In the language of the previous section, π is a Markov field if and only if it is positive and its dependency graph is a subgraph of G. The key result about such measures is their representation in terms of a ‘potential function’ φ, in a form known as a ‘Gibbs state’. Recall the set K of cliques of the graph G. (7.9) Definition. A probability measure π on 6 is called a Gibbs state if there exists a ‘potential’ function φ : 2 V → R, satisfying φC = 0 if C ∈ / K, such that X  (7.10) π(B) = exp φK , B ⊆ V. K ⊆B

We allow the empty set in the above summation, so that log π(∅) = φ ∅ . Gibbs states are thus named after Josiah Willard Gibbs, whose volume [87] made available the foundations of statistical mechanics. A simplistic motivation for the form of (7.10) is as follows. Suppose that each state σ P has an energy E σ , and a probability π(σ ). We constrain the average energy E = σ E σ π(σ ) to be fixed, and we maximize the entropy X η(π ) = − π(σ ) log2 π(σ ). σ ∈6

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Markov fields and Gibbs states

129

With the aid of a Lagrange multiplier β, we find that π(σ ) ∝ e−β E σ ,

σ ∈ 6.

The theory of thermodynamics leads to the expression β = 1/(kT ) where k is Boltzmann’s constant and T is (absolute) temperature. Formula (7.10) arises when the energy E σ may be expressed as the sum of the energies of the sub-systems indexed by cliques. (7.11) Theorem. A positive probability measure π on 6 is a Markov field if and only if it is a Gibbs state. The potential function φ corresponding to the Markov field π is given by X φK = (−1)|K \L| log π(L), K ∈ K. L⊆K

A positive probability measure π is said to have the local Markov property if it satisfies the global property (7.8) for all singleton sets W and all s ∈ 6. The global property evidently implies the local property, and it turns out that the two properties are equivalent. For notational convenience, we denote a singleton set {w} as w. (7.12) Proposition. Let π be a positive probability measure on 6. The following three statements are equivalent: (i) π satisfies the global Markov property, (ii) π satisfies the local Markov property, (iii) for all A ⊆ V and any pair u, v ∈ V with u ∈ / A, v ∈ A and u  v, (7.13)

π( A ∪ u \ v) π( A ∪ u) = . π( A) π( A \ v)

Proof. First, assume (i), so that (ii) is implied trivially. Let u ∈ / A, v ∈ A, and u  v. Applying (7.8) with W = {u} and, for w 6= u, sw = 1 if and only if w ∈ A, we find that (7.14)

π( A ∪ u) = π(σu = 1 | σV \u = A) π( A) + π( A ∪ u) = π(σu = 1 | σ1u = A ∩ 1u) = π(σu = 1 | σV \u = A \ v) =

π( A ∪ u \ v) . π( A \ v) + π( A ∪ u \ v)

since v ∈ / 1u

Equation (7.14) is equivalent to (7.13), whence (ii) and (iii) are equivalent under (i). c G. R. Grimmett 6 February 2009

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Gibbs States

It remains to show that the local property implies the global property. The proof requires a short calculation, and may be done either by Theorem 7.1 or within the proof of Theorem 7.11. We follow the first route here. Assume that π is positive and satisfies the local Markov property. Then u ⊥ v for all u, v ∈ V with u  v. By Theorem 7.1, there exist functions f K , K ∈ M, such that (7.15)

π( A) =

Y

K ∈M

f K ( A ∩ K ),

A ⊆ V.

Let W ⊆ V . By (7.15), for A ⊆ W and C ⊆ V \ W , Q K ∈M f K (( A ∪ C) ∩ K ) Q π(σW = A | σV \W = C) = P . B⊆W K ∈M f K ((B ∪ C) ∩ K )

Any clique K with K ∩ W = ∅ makes the same contribution f K (C ∩ K ) to both numerator and denominator, and may be cancelled. The remaining cliques are b = W ∪ 1W , so that subsets of W Q K ∈M, K ⊆W f K (( A ∪ C) ∩ K ) Q . π(σW = A | σV \W = C) = P B⊆W K ∈M, K ⊆W f K ((B ∪ C) ∩ K )

The right side does not depend on σ V \W , whence

π(σW = A | σV \W = C) = π(σW = A | σ1W = C ∩ 1W ) as required for the global Markov property.



Proof of Theorem 7.11. Assume first that π is a positive Markov field, and let (7.16)

φC =

X

(−1)|C\L| log π(L),

C ⊆ V.

L⊆C

By the inclusion–exclusion principle, log π(B) =

X

C⊆B

φC ,

B ⊆ V,

and we need only show that φC = 0 for C ∈ / K. Suppose u, v ∈ C and u  v. By (7.16), φC =

X

L⊆C\{u,v}

π(L ∪ u ∪ v) (−1)|C\L| log π(L ∪ u)

,

! π(L ∪ v) , π(L)

which equals zero by the local Markov property and Proposition 7.12. Therefore, π is a Gibbs state with potential function φ. c G. R. Grimmett 6 February 2009

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Ising and Potts models

131

Conversely, suppose that π is a Gibbs state with potential function φ. Evidently, π is positive. Let A ⊆ V , and u ∈ / A, v ∈ A with u  v. By (7.10), 

π( A ∪ u) log π( A)



= =

X

φK

K ⊆A∪u, u∈K

X

K ⊆A∪u\v, u∈K

φK

since u  v and K ∈ K



 π( A ∪ u \ v) = log . π( A \ v) The claim follows by Proposition 7.12.



We close this section with some notes on the history of Theorem 7.11. It may be derived from Brook’s theorem, Theorem 7.1, but it is perhaps more informative to prove it directly as above via the inclusion–exclusion principle. It is normally attributed to Hammersley and Clifford, and it was circulated (with a more complicated formulation and proof) in an unpublished note of 1971, [115] (see [63]). Versions of the theorem may be found in the later work of several authors. The above proof is taken from [92], the author’s earliest published paper and part of his 1972 MSc dissertation at Oxford University. The assumption of positivity is important, and complications arise for non-positive measures, see [166]. For applications of the Gibbs/Markov equivalence in statistics, see, for example, [142].

7.3 Ising and Potts models In a famous experiment, a piece of iron is exposed to a magnetic field. The field is increased from zero to a maximum, and then diminished to zero. If the temperature is sufficiently low, the iron retains some residual magnetization, otherwise it does not. There is a critical temperature for this phenomenon, often named the Curie point after Pierre Curie, who reported this discovery in his 1895 thesis. The famous (Lenz–)Ising model for such ferromagnetism, [127], may be summarized as follows. Let particles be positioned at the points of some lattice in Euclidean space. Each particle may be in either of two states, representing the physical states of ‘spin-up’ and ‘spin-down’. Spin-values are chosen at random according to a Gibbs state governed by interactions between neighbouring particles, and given in the following way. Let G = (V, E) be a finite graph representing part of the lattice. Each vertex x ∈ V is considered as being occupied by a particle that has a random spin. Spins are assumed to come in two basic types (‘up’ and ‘down’), and thus we take the set 6 = {−1, +1}V as the sample space. The appropriate probability mass function λβ,J,h on 6 has three parameters satisfying β, J ∈ [0, ∞) and h ∈ R, and is given c G. R. Grimmett 6 February 2009

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[7.3]

Gibbs States

by (7.17)

λβ,J,h (σ ) =

1 −β H (σ ) e , ZI

σ ∈ 6,

where the ‘Hamiltonian’ H : 6 → R and the ‘partition function’ Z I are given by (7.18)

H (σ ) = −J

X

e=hx,yi∈E

σx σ y − h

X

ZI =

σx ,

x∈V

X

e−β H (σ ) .

σ ∈6

The physical interpretation of β is as the reciprocal 1/T of temperature, of J as the strength of interaction between neighbours, and of h as the external magnetic field. We shall consider here only the case of zero external-field, and we assume henceforth that h = 0. Each edge has equal interaction strength J in the above formulation. Since β and J occur only as a product β J , the measure λβ,J,0 has effectively only a single parameter β J . In a more complicated measure not studied here, different edges e are permitted to have different interaction strengths Je . In the meantime we shall set J = 1, and write λβ = λβ,1,0 Whereas the Ising model permits only two possible spin-values at each vertex, the so-called (Domb–)Potts model [177] has a general number q ≥ 2, and is governed by the following probability measure. Let q be an integer satisfying q ≥ 2, and take as sample space the set of vectors 6 = {1, 2, . . . , q}V . Thus each vertex of G may be in any of q states. For an edge e = hx, yi and a configuration σ = (σ x : x ∈ V ) ∈ 6, we write δe (σ ) = δσx ,σ y where δi, j is the Kronecker delta. The relevant probability measure is given by (7.19)

πβ,q (σ ) =

1 −β H 0 (σ ) e , ZP

σ ∈ 6,

where Z P = Z P (β, q) is the appropriate partition function (or normalizing constant) and the Hamiltonian H 0 is given by (7.20)

H 0 (σ ) = −

X

δe (σ ).

e=hx,yi∈E

In the special case q = 2, (7.21)

δσx ,σ y = 12 (1 + σx σ y ),

σx , σ y ∈ {−1, +1},

It is easy to see in this case that the ensuing Potts model is simply the Ising model with an adjusted value of β, in that πβ,2 is the measure obtained from λβ/2 by re-labelling the local states. We mention one further generalization of the Ising model, namely the so-called n-vector or O(n) model. Let n ∈ {1, 2, . . . } and let S n−1 be the set of vectors of c G. R. Grimmett 6 February 2009

[7.4]

Exercises

133

Rn with unit length, that is, the (n − 1)-sphere. A ‘model’ is said to have O(n)

symmetry if its Hamiltonian is invariant under the operation on S n−1 of n × n orthonormal matrices. One such model is the n-vector model on G = (V, E), with Hamiltonian X Hn (s) = − sx · s y , s = (sv : v ∈ V ) ∈ (S n−1 )V , e=hx,yi∈E

where sx · s y denotes the dot product. When n = 1, this is simply the Ising model. It is called the X/Y model when n = 2, and the Heisenberg model when n = 3. The Ising and Potts models have very rich theories, and are amongst the most intensively studied of models of statistical mechanics. In ‘classical’ work, they are studied via cluster expansions and correlation inequalities. The so-called ‘randomcluster model’, developed by Fortuin and Kasteleyn around 1960, provides a single framework that incorporates the percolation, Ising, and Potts models, as well as electrical networks, uniform spanning trees and forests. It enables a representation of the two-point correlation function of a Potts model as a connection probability of an appropriate model of stochastic geometry, and this in turn allows the use of geometrical techniques already refined in the case of percolation. The randomcluster model is defined and described in Chapter 8, see also [98]. The q = 2 Potts model is of course the Ising model, and special features of the number 2 allow a special analysis for the Ising model not yet replicated for general Potts models. This method is termed the ‘random-current representation’, and it has been especially fruitful in the study of the phase transition of the Ising model on Ld . See [3, 7, 10] and [98, Chap. 9].

7.4 Exercises 7.1. [166] Investigate the Gibbs/Markov equivalence for probability measures that have zeroes. 7.2. Ising model. Let G = (V, E) be a finite graph, and let λ be the probability measure on 6 = {−1, +1} V given by 

λ(σ ) ∝ exp β

X

e=hi, j i



σi σ j ,

σ ∈ 6,

where β > 0. Thinking of 6 as a partially ordered set (where σ ≤ σ 0 if and only if σi ≤ σi0 for all i ), show that: (a) for v ∈ V , λ(· | σv = −1) ≤st λ ≤st λ(· | σv = +1), (b) λ satisfies the FKG lattice condition, and hence is positively associated.

c G. R. Grimmett 6 February 2009

8 Random-Cluster Model

The basic properties of the model are summarized, and its relationship to the Ising and Potts models described. The phase transition is defined in terms of the infinite-volume measures. After an account of a number of areas meritorious of further research, there is a section devoted to planar duality and the conjectured value of the critical point on the square lattice. The random-cluster model is linked in more than one way to the study of a random even subgraph of a graph.

8.1 The random-cluster and Ising/Potts models Let G = (V, E) be a finite graph, and write  = {0, 1} E . For ω ∈ , we write η(ω) = {e ∈ E : ω(e) = 1} for the set of open edges, and k(ω) for the number of connected components1 , or ‘clusters’, of the subgraph (V, η(ω)). The randomcluster measure on , with parameters p ∈ [0, 1], q ∈ (0, ∞) is the probability measure given by   1 Y ω(e) 1−ω(e) p (1 − p) q k(ω) , ω ∈ , (8.1) φ p,q (ω) = Z e∈E

where Z = Z G, p,q is the normalizing constant. Some history. This measure was introduced by Fortuin and Kasteleyn in a series of papers dated around 1970. They sought a unification of the theory of electrical networks, percolation, Ising, and Potts models, and were motivated by the observation that each of these systems satisfies a certain series/parallel law. Percolation is evidently retrieved by setting q = 1, and it turns out that electrical networks arise via the UST limit obtained on taking the limit p, q → 0 in such a way that q/ p → 0. The relationship to Ising/Potts models is more interesting in that it involves a transformation of measures described next. In brief, connection probabilities for the random-cluster measure correspond to correlations 1 It

is important to include isolated vertices in this count.

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The random-cluster and Ising/Potts models

135

for ferromagnetic Ising/Potts models, and this allows a geometrical interpretation of their correlation structure. A fuller account of the random-cluster model and its history and associations may be found in [98]. When the emphasis is upon its connection to Ising/Potts models, the random-cluster model is often called the ‘FK representation’. In the remainder of this section, we summarize the relationship between a Potts model on G = (V, E) with an integer number q of local states, and the random-cluster measure φ p,q . As configuration space for the Potts model, we take 6 = {1, 2, . . . , q}V . Let F be the subset of the product space 6 ×  containing all pairs (σ, ω) such that: for every edge e = hx, yi ∈ E, if ω(e) = 1 then σx = σ y . That is, F contains all pairs (σ, ω) such that σ is constant on each cluster of ω. Let φ p = φ p,1 be product measure on ω with density p, and let µ be the probability measure on 6 ×  given by (8.2)

µ(σ, ω) ∝ φ p (ω)1 F (σ, ω),

(σ, ω) ∈ 6 × ,

where 1 F is the indicator function of F. Four calculations are now required, in order to determine the two marginal measures of µ and the two conditional measures. It turns out that the two marginals are exactly the q-state Potts measure on 6 (with suitable pair-interaction) and the random-cluster measure φ p,q . Marginal on 6. When we sum µ(σ, ω) over ω ∈ , we have a free choice except in that ω(e) = 0 whenever σ x 6= σ y . That is, if σx = σ y , there is no constraint on the local state ω(e) of the edge e = hx, yi; the sum for this edge is simply p + (1 − p) = 1. We are left with edges e with σ x 6= σ y , and therefore (8.3)

µ(σ, ·) :=

X

ω∈

µ(σ, ω) ∝

Y

e∈E

(1 − p)1−δe (σ ) ,

where δe (σ ) is the Kronecker delta (8.4)

δe (σ ) = δσx ,σ y

e = hx, yi ∈ E.

Otherwise expressed,  X  µ(σ, ·) ∝ exp β δe (σ ) , e∈E

σ ∈ 6,

where (8.5)

p = 1 − e−β .

This is the Potts measure πβ,q of (7.19). Note that β ≥ 0, which is to say that the model is ferromagnetic. c G. R. Grimmett 6 February 2009

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[8.2]

Random-Cluster Model

Marginal on . For given ω, the constraint on σ is that it be constant on open clusters. There are q k(ω) such spin configurations, and µ(σ, ω) is constant on this set. Therefore, µ(·, ω) :=

X

σ ∈6

µ(σ, ω) ∝

∝ φ p,q (ω),

Y

p

ω(e)

e∈E

(1 − p)

1−ω(e)

 q k(ω)

ω ∈ .

The conditional measures. Given ω, the conditional measure on 6 is obtained by putting (uniformly) random spins on entire clusters of ω, constant on given clusters, and independent between clusters. Given σ , the conditional measure on  is obtained by setting ω(e) = 0 if δe (σ ) = 0, and otherwise ω(e) = 1 with probability p (independently of other edges). The ‘two-point correlation function’ of the Potts measure πβ,q on G = (V, E) is the function τβ,q given by τβ,q (x, y) = πβ,q (σx = σ y ) −

1 , q

x, y ∈ V.

The ‘two-point connectivity function’ of the random-cluster measure φ p,q is the probability φ p,q (x ↔ y) of an open path from x to y. It turns out that these ‘two-point functions’ are (except for a constant factor) the same. (8.6) Theorem [133]. For q ∈ {2, 3, . . . }, β ≥ 0, and p = 1 − e −β , τβ,q (x, y) = (1 − q −1 )φ p,q (x ↔ y). Proof. We work with the conditional measure µ(σ | ω) thus: τβ,q (x, y) = = =

X σ,ω

X

 1{σx =σ y } (σ ) − q −1 µ(σ, ω)

φ p,q (ω)

ω

X ω

X



σ

  µ(σ | ω) 1{σx =σ y } (σ ) − q −1

φ p,q (ω) (1 − q −1 )1{x↔y} (ω) + 0 · 1{x ↔y} / (ω)

= (1 − q −1 )φ p,q (x ↔ y), and the claim is proved.

c G. R. Grimmett 6 February 2009

 

[8.3]

Infinite-volume limits and phase transition

137

8.2 Basic properties We list some of the fundamental properties of random-cluster measures in this section. (8.7) Theorem. The measure φ p,q satisfies the FKG lattice condition if q ≥ 1, and is thus positively associated. Proof. If p = 0, 1, the conclusion is obvious. Assume 0 < p < 1, and check the FKG lattice condition (4.12), which amounts to the assertion that k(ω ∨ ω0 ) + k(ω ∧ ω0 ) ≥ k(ω) + k(ω0 ),

ω, ω0 ∈ .

This is left as a graph-theoretic exercise for the reader.



(8.8) Theorem. Comparison inequalities [81]. We have that (8.9)

φ p0 ,q 0 ≤st φ p,q

if

(8.10)

φ p0 ,q 0 ≥st φ p,q

if

p 0 ≤ p, q 0 ≥ q, q 0 ≥ 1,

p p0 ≥ , q 0 ≥ q, q 0 ≥ 1. 0 0 q (1 − p ) q(1 − p)

Proof. This follows by the Holley inequality, Theorem 4.4, on checking condition (4.5).  In the next theorem, the role of the graph G is emphasized in the use of the notation φG, p,q . The graph G\e (respectively, G.e) is obtained from G by deleting (respectively, contracting) the edge e. (8.11) Theorem [81]. Let e ∈ E. (a) Conditional on ω(e) = 0, the measure obtained from φ G, p,q is φG\e, p,q . (b) Conditional on ω(e) = 1, the measure obtained from φ G, p,q is φG.e, p,q . Proof. This is an elementary calculation of conditional probabilities.



Maybe mention UST, and negative association/disjoint occurrence.

8.3 Infinite-volume limits and phase transition Let d ≥ 2, and  = {0, 1}E . The appropriate σ -field of  is the σ -field F generated by the finite-dimensional sets. Let 3 be a finite box in Zd . For b ∈ {0, 1} define b3 = {ω ∈  : ω(e) = b for e ∈ / E3 }, d

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[8.3]

Random-Cluster Model

where E A is the set of edges of Ld joining pairs of vertices belonging to A. On b b3 we define a random-cluster measure φ3, p,q as follows. For p ∈ [0, 1] and q ∈ (0, ∞), let b (8.12) φ3, p,q (ω) =

1 b Z 3, p,q

(

Y

e∈E3

)

p ω(e) (1 − p)1−ω(e) q k(ω,3) ,

ω ∈ b3 ,

where k(ω, 3) is the number of clusters of (Zd , η(ω)) that intersect 3 (here, as before, η(ω) = {e ∈ Ed : ω(e) = 1} is the set of open edges). The boundary condition b = 0 (respectively, b = 1) is sometimes termed ‘free’ (respectively, ‘wired’). (8.13) Theorem [93]. Let q ≥ 1. The weak limits b φ bp,q = lim φ3, p,q , 3→Zd

b = 0, 1,

exist, and are translation-invariant and ergodic. Proof. Let A be an increasing cylinder event defined in terms of the edges lying in some finite set S. If 3 ⊆ 30 and 3 includes the ‘base’ S of the cylinder A, 1 1 1 φ3, p,q ( A) = φ30 , p,q ( A | all edges in E30 \3 are open) ≥ φ30 , p,q ( A),

where we have used Theorem 8.11 and the FKG inequality. Therefore, the limit 1 lim3→Zd φ3, p,q ( A) exists by monotonicity. Since F is generated by such events A, the weak limit φ 1p,q exists. A similar argument is valid in the case b = 0. Translation-invariance holds in very much the same way as in the proof of Theorem 2.10. The proof of ergodicity is deferred to Exercises 8.9–8.10.  The measures φ 0p,q and φ 1p,q are called ‘random-cluster measures’ on Ld with parameters p and q, and they are extremal in the following sense. One may generate ostensibly larger families of infinite-volume random-cluster measures ξ by either of two routes. In the first, one considers measures φ3, p,q on E3 with more general boundary conditions ξ , in order to construct a set W p,q of ‘weaklimit random-cluster measures’. The second uses a type of Dobrushin–Lanford– Ruelle (DLR) formalism rather than weak limits (see [93] and [98, Chap. 4]). More precisely, one considers measures µ on (, F ) whose measure on any box 3, conditional on the state ξ off 3, is the conditional random-cluster measure ξ φ3, p,q . Such a µ is called a ‘DLR random-cluster measure’, and we write R p,q for the set of DLR measures. The relationship between W p,q and R p,q is not fully understood, and we make one remark about this. Any element µ of the closed convex hull of W p,q with the so-called ‘0/1-infinite-cluster property’ (that is, µ(I ∈ {0, 1}) = 1 where I is the number of infinite open clusters) belongs to R p,q , see [98, Sect. 4.4]. The standard way of showing the 0/1-infinite-cluster c G. R. Grimmett 6 February 2009

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139

property is via the Burton–Keane argument used in the proof of Theorem 5.22. One may show, in particular, that φ 0p,q , φ 1p,q ∈ R p,q .

Henceforth we assume that q ≥ 1. The measures φ 0p,q and φ 1p,q are extremal in the sense that (8.14)

φ 0p,q ≤st φ p,q ≤st φ 1p,q ,

φ p,q ∈ W p,q ∪ R p,q ,

whence there exists a unique random-cluster measure (in either of the above senses) if and only if φ 0p,q = φ 1p,q . It is a general fact that such extremal measures are invariably ergodic, see [86, 98]. Turning to the question of phase transition, and remembering percolation, we define the percolation probabilities (8.15)

θ b ( p, q) = φ bp,q (0 ↔ ∞),

b = 0, 1,

that is, the probability that 0 belongs to an infinite open cluster. The corresponding critical values are given by (8.16)

pcb (q) = sup{ p : θ b ( p, q) = 0},

b = 0, 1.

Faced possibly with two (or more) distinct critical values, we present the following result. (8.17) Theorem [9, 93]. Let d ≥ 2 and q ≥ 1. We have that: (i) φ 0p,q = φ 1p,q if θ 1 ( p, q) = 0, (ii) there exists a countable subset Dd,q of [0, 1], possibly empty, such that φ 0p,q = φ 1p,q if and only if p ∈ / Dd,q .

Sketch proof. The argument for (i) is as follows. Clearly, (8.18)

θ 1 ( p, q) = lim φ 1p,q (0 ↔ ∂3). 3↑Zd

Suppose θ 1 ( p, q) = 0, and consider a large box 3 with 0 in its interior. On building the clusters that intersect the boundary ∂3, with high probability we do not reach 0. That is, with high probability, there exists a ‘cut-surface’ S between 0 and ∂3 that comprises only closed edges. The position of S may be taken to be measurable on its exterior, whence the conditional measure on the interior of S is a free random-cluster measure. Passing to the limit as 3 ↑ Zd , we find that the free and wired measures are equal. The argument for (ii) is based on a classical method of statistical mechanics using convexity. Let Z G, p,q be the partition function of the random-cluster model on a graph G = (V, E), and set X YG, p,q = (1 − p)−|E | Z G, p,q = eπ |η(ω)| q k(ω) , ω∈{0,1} E

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where π = log[ p/(1 − p)]. It is easily seen that log Y G, p,q is a convex function of π . By a standard method based on the negligibility of the boundary of a large box 3 compared with its volume, the limit ‘pressure function’ 5(π, q) = lim

3↑Zd



1 ξ log Y3, p,q |E 3 |



exists and is independent of the boundary configuration ξ ∈ . Since 5 is the limit of convex functions of π , it is convex, and hence differentiable except on some countable set D of values of π . Furthermore, for π ∈ / D, the derivative of ξ −1 |E3 | log Y3, p,q converges to that of 5. The former derivative may be interpreted in terms of the edge-density of the measures, and therefore the limits of the last are independent of ξ for any π at which 5(π, q) is differentiable.2 Uniqueness of random-cluster measures follows by (8.14) and stochastic ordering: if µ 1 , µ2 are probability measures on (, F ) with µ1 ≤st µ2 and satisfying µ1 (e is open) = µ2 (e is open),

e ∈ E,

then3 µ1 = µ2 .



By Theorem 8.17, θ 0 ( p, q) = θ 1 ( p, q) for p ∈ / Dd,q , whence pc0 (q) = pc1 (q). Henceforth we refer to the critical value as pc = pc (q). The following is an important conjecture. (8.19) Conjecture. There exists Q = Q(d) such that: (i) if q < Q(d), then θ 1 ( pc , q) = 0 and Dd,q = ∅,

(ii) if q > Q(d), then θ 1 ( pc , q) > 0 and Dd,q = { pc }.

Reference physics literature for q ≥ 4, d = 2. In the physical vernacular, there is conjectured a critical value of q beneath which the phase transition is continuous (‘second order’) and above which it is discontinuous (‘first order’). Following work of Roman Kotecky´ and Senya Shlosman [139], it was proved in [140] that there is a first-order transition for large q, see [98, Sects 6.4, 7.5]. It is expected that Q(d) =



4 2

if d = 2,

if d ≥ 6.

This may be contrasted with the best current estimate in two dimensions, namely Q(2) ≤ 25.72, see [98, Sect. 6.4]. It is a basic fact that pc (q) is non-trivial. 2 Expand

3 Exercise.

Recall Strassen’s Theorem 4.2.

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(8.20) Theorem [9]. If d ≥ 2 and q ≥ 1 then 0 < pc (q) < 1. It is an open problem to find a satisfactory definition of pc (q) for q < 1, although it may be shown by the comparison inequalities (Theorem 8.8) that there is no infinite cluster for q ∈ (0, 1) and small p, and conversely there is an infinite cluster for q ∈ (0, 1) and large p.

Proof. Let q ≥ 1. By Theorem 8.8, φ 1p0 ,1 ≤st φ 1p,q ≤st φ p,1 , where p 0 = p/[ p + q(1 − p)]. We apply this inequality to the increasing event {0 ↔ ∂3}, and let 3 ↑ Zd to obtain via (8.22) that (8.21)

pc (1) ≤ pc (q) ≤

q pc (1) , 1 + (q − 1) pc (1)

q ≥ 1,

where 0 < pc (1) < 1 by Theorem 3.2.



Finally, we review the relationship between the random-cluster and Potts phase transitions. The ‘order parameter’ of the Potts model is the ‘magnetization’ given by   1 1 , M(β, q) = lim π3,β (σ0 = 1) − q 3→Zd

1 is the Potts measure on 3 ‘with boundary condition 1’. We may think where π3,β of M(β, q) as a measure of the degree to which the boundary condition ‘1’ is noticed at the origin after taking the infinite-volume limit. By an application of Theorem 8.6 to a suitable graph obtained from 3, 1 (σ0 = 1) − π3,q

1 1 = (1 − q −1 )φ3, p,q (0 ↔ ∂3) q

where p = 1 − e−β . It may be deduced 4 that 1 θ 1 ( p, q) = lim φ3, p,q (0 ↔ ∂3).

(8.22)

3↑Zd

Therefore 1 M(β, q) = (1 − q −1 ) lim φ3, p,q (0 ↔ ∂3)

= (1 − q

−1

3→Zd 1

)θ ( p, q),

by (8.22). That is, M(β, q) and θ 1 ( p, q) differ by the factor 1 − q −1 .

4 Exercise

8.8.

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8.4 Open problems Many questions remain at least partly unanswered for the random-cluster model, and we list a few of these here. Further details may be found in [98]. A. The case q < 1. Less is known when q < 1 owing to the failure of the FKG inequality. A possibly optimistic conjecture is that some version of negative association holds when q < 1, and this might imply the existence of infinite-volume limits. One may use comparison arguments to study infinite-volume randomcluster measures for sufficiently small or large p, but there is no proof of the existence of a unique point of phase transition. A certain amount is known in the limit as q → 0, depending on how p behaves in this limit. BK? The case q < 1 is of more mathematical than physical interest, although the various limits as q → 0 are relevant to the theory of algorithms and complexity. Henceforth, we assume q ≥ 1. B.

Exponential decay. Prove that


φ p,q 0 ↔ ∂[−n, n]d ≤ e−αn ,

n ≥ 1,

for some α = α( p, q) satisfying α > 0 when p < pc (q). This has been proved for sufficiently small values of p, but no proof is known (for general q and any given d ≥ 2) right up to the critical point. The case q = 2 is special, since this corresponds to the Ising model, for which the random-current representation has allowed a rich theory, see [98, Sect. 9.3]. Exponential decay is proved to hold for general d, when q = 2, and also for sufficiently large q (see D below). C. Uniqueness of random-cluster measures. Prove all or part of Conjecture 8.19. That is, show that φ 0p,q = φ 1p,q for p 6= pc (q). And, furthermore, that uniqueness holds when p = pc (q) if and only if q is sufficiently small. These statements are trivial when q = 1, and uniqueness is proved when q = 2 and p 6= pc (2), using the theory of the Ising model alluded to above. The situation is curious when q = 2 and p = pc (2), in that uniqueness is proved so long as d 6= 3, see [98, Sect. 9.4]. When q is sufficiently large, it is known as in D below that there is a unique random-cluster measure when p 6= pc (q) and a multiplicity of such measures when p = pc (q).

D. First/second order phase transition. Much interest in Potts and randomcluster measures is focussed on the fact that nature of the phase transition depends on whether q is small or large, see for example Conjecture 8.19. For small q, the singularity is expected to be continuous and of power type. For large q, there is a discontinuity in the order parameter θ 1 (·, q), and a ‘mass gap’ at the critical point (that is, when p = pc (q), the φ 0p,q -probability of a long path decays exponentially, while the φ 1p,q -probability is bounded away from 0). c G. R. Grimmett 6 February 2009

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143

Of the possible questions, we ask for a proof of the existence of a value Q = Q(d) separating the second- from the first-order transition. E. Slab critical point. It was important for supercritical percolation in three and more dimensions to show that percolation in Ld implies percolation in a sufficiently fat ‘slab’, see Theorem 5.17. A version of the corresponding problem for the random-cluster model is as follows. Let q ≥ 1 and d ≥ 3, and write S(L , n) for the ‘slab’ S(L , n) = [0, L − 1] × [−n, n]d−1 . 0 Let ψ L ,n, p,q = φ S(L ,n), p,q be the random-cluster measure on S(L , n) with parameters p, q, and free boundary conditions. Write 5( p, L) for the property that:

∃α > 0 such that: ∀n ≥ 1, ∀x ∈ S(L , n), ψ L ,n, p,q (0 ↔ x) > α. It is not hard to see that 5( p, L) ⇒ 5( p 0 , L 0 ) if p ≤ p 0 and L ≤ L 0 , and it is thus natural to define  (8.23) b pc (q, L) = inf p : 5( p, L) occurs , b pc (q) = lim b pc (q, L). L→∞

5

Clearly, pc (q) ≤ b pc (q) < 1. It is believed that equality holds in that b pc (q) = pc (q), and it is a major open problem to prove this. A positive resolution would have implications for the exponential decay of truncated cluster-sizes, and for the existence of a Wulff crystal for all p > pc (q) and q ≥ 1. See Figure 5.3 and [55, 56, 57].

F. Roughening transition. While it is believed that there is a unique randomcluster measure except possibly at the critical point, there can exist a multitude of random-cluster-type measures with the striking property of non-translationinvariance. Take a box 3n = [−n, n]d in d ≥ 3 dimensions (the following construction fails in 2 dimensions). We may think of ∂3n as comprising a northern and southern hemisphere, with the ‘equator’ {x ∈ ∂3n : x d = 0} as interface. Let φ n, p,q be the random-cluster measure on 3n with a wired boundary condition on the northern (respectively, southern) hemisphere and conditioned on the event that no open path joins a point of the northern to a point of the southern hemisphere. By the compactness of , the sequence (φ n, p,q : n ≥ 1) possesses weak limits. Let φ p,q be such a weak limit. It is a geometrical fact that, in any configuration ω on 3n , there exists an interface I (ω) separating the points joined to the northern hemisphere from those joined to the southern hemisphere. This interface passes around the equator, and its closest point to the origin is at some distance Hn , say. It may be shown that, for q ≥ 1 and sufficiently large p, the laws of the Hn are tight, whence the weak limit 5 Use

FKG, explain

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φ p,q is not translation-invariant. Such measures are termed ‘Dobrushin measures’ after their discovery for the Ising model in [64]. There remain two important questions. Firstly, for d ≥ 3 and q ≥ 1, does there exist a value e p (q) such that Dobrushin measures exist for p > e p (q) and not for p< e p (q)? And secondly, for what dimensions d do Dobrushin measures exist for all p > pc (q)? A fuller account may be found in [98, Chap. 7].

G. In two dimensions. There remain some intriguing conjectures in the playground of the square lattice L2 , and some of these are described in the next section.

8.5 In two dimensions Consider the special case of the square lattice L2 . Random-cluster measures on L2 have a property of self-duality that generalizes that of bond percolation. (We recall the discussion of duality after equation (3.7).) The most provocative conjecture is that the critical point equals the so-called self-dual point. (8.24) Conjecture. For d = 2 and q ≥ 1, pc (q) =

√

q √ . 1+ q

This formula is proved rigorously when q = 1 (percolation), when q = 2 (Ising model), and for sufficiently large values of q (namely q ≥ 25.72). The conjecture is motivated as follows. Let G = (V, E) be a finite planar graph, and G d = (Vd , E d ) its dual graph. To each ω ∈  = {0, 1} E , there corresponds the dual configuration ωd ∈ d = {0, 1} E d , given by ωd (ed ) = 1 − ω(e),

e ∈ E.

(Note that this definition of the dual configuration differs from that used in Chapter 3 for percolation.) By drawing a picture, one may become convinced that every face of (V, η(ω)) contains a unique component of (Vd , η(ωd )), and therefore the number f (ω) of faces (including the infinite face) of (V, η(ω)) satisfies (8.25)

f (ω) = k(ωd ).

The random-cluster measure on G satisfies |η(ω)|  p q k(ω) . φG, p,q (ω) ∝ 1− p Using (8.25), Euler’s formula, (8.26)

k(ω) = |V | − |η(ω)| + f (ω) − 1,

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145

and the fact that |η(ω)| + |η(ωd )| = |E|, we have that   q(1 − p) |η(ωd )| k(ωd ) q , φG, p,q (ω) ∝ p which is to say that (8.27) where (8.28)

φG, p,q (ω) = φG d , pd ,q (ωd ),

ω ∈ ,

pd q(1 − p) = . 1 − pd p

The unique fixed point of the mapping p 7→ pd is given by p = κq where κ is the ‘self-dual point’ √ q κq = √ . 1+ q Turning to the square lattice, let G = 3 = [0, n]2 , with dual graph G d = 3d obtained from the box [−1, n]2 + ( 12 , 21 ) by identifying all boundary vertices. By (8.27), (8.29)

0 1 φ3, p,q (ω) = φ3d , pd ,q (ωd )

for configurations ω on 3 (and with a small ‘fix’ on the boundary of 3 d ). Letting n → ∞, we obtain that (8.30)

φ 0p,q ( A) = φ 1pd ,q ( Ad )

for all cylinder events A, where A d = {ωd : ω ∈ A}. The duality relation (8.30) is useful, especially if p = pd = κq . In particular, the proof that θ ( 21 ) = 0 for percolation (see Theorem 5.33) may be adapted to obtain θ 0 (κq , q) = 0,

(8.31) whence (8.32)

√ q pc (q) ≥ √ , 1+ q

q ≥ 1.

In order to obtain the formula of Conjecture 8.24, it would be enough to show that,
A φ 0p,q 0 ↔ ∂[−n, n]2 ≤ , n ≥ 1, n where A = A( p, q) < ∞ for p < κq . See [89, 98]. The case q = 2 is very special, because it is related to the Ising model, for which there is a rich and exact theory going back to Onsager [168]. As an illustration of this connection in action, we include a proof that the wired random-cluster measure has no infinite cluster at the self-dual point. The corresponding conclusion in believed to hold if and only if q ≤ 4, but a full proof is elusive. c G. R. Grimmett 6 February 2009

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(8.33) Theorem. For d = 2, θ 1 (κ2 , 2) = 0.

Proof. Of the several proofs of this statement, we summarise the recent simple proof of Werner [210]. Let q = 2, and write φ b = φ bpsd (q),q . Let ω ∈  be a configuration of the random-cluster model sampled according to φ 0 . To each open cluster of ω we allocate the spin +1 with probability 21 , and −1 otherwise. Thus, spins are constant within clusters, and independent between clusters. Let σ be the resulting spin configuration, and let µ0 be its law. We do the same with ω sampled from φ 1 , with the difference that any infinite cluster is allocated the spin +1. It is not hard to see6 that the resulting measure µ1 is the infinite-volume Ising measure with boundary condition +1. The spin-space 2 6 = {−1, +1}Z is a partially ordered set, and it may be checked using the Holley inequality7 , Theorem 4.4, and passing to an infinite-volume limit that µ0 ≤st µ1 .

(8.34)

We shall be interested in two notions of connectivity in Z2 , the first of which is the usual one, denoted ↔. If we add both diagonals to each face of Z 2 , we obtain a new graph with so-called ∗-connectivity relation denoted ↔∗ . A cycle in this new graph is called a ∗-cycle. Each σ ∈ 6 amounts to a partition of Z2 into maximal clusters with constant spin. A cluster labelled +1 (respectively, −1) is called a (+)-cluster (respectively, (−)-cluster). Let N + (σ ) (respectively, N − (σ )) be the number of infinite (+)clusters (respectively, (−)-clusters). By (8.31), φ 0 (0 ↔ ∞) = 0, whence, by Exercise 8.14, µ0 is ergodic. One may apply the Burton–Keane argument of Section 5.3 to deduce that: either

µ0 (N + = 1) = 1 or

µ0 (N + = 0) = 1.

One may now use Zhang’s argument (as in the proof of (8.31) and Theorem 5.33), and the fact that N + and N − have the same law, to deduce that µ0 (N + = 0) = µ0 (N − = 0) = 1.

(8.35)

Let A be an increasing cylinder event of 6 defined in terms of states of vertices in some box 3. By (8.35), there are (µ0 -a.s.) no infinite (−)-clusters intersecting 3, so that 3 lies in the interior of some ∗-cycle labelled +1. let 3n = [−n, n]2 with n large, and let Dn be the event that 3n contains a ∗-cycle labelled +1 with 3 in its interior. By the above, µ0 (Dn ) → 1 as n → ∞. The event Dn is an increasing subset of 6, whence, by (8.34), µ1 (Dn ) → 1 6 This

7 See

as n → ∞.

is formalized in [98, Sect. 4.6]; see also Exercise 8.14. Exercise 7.2.

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On Dn , we find the ‘outermost’ ∗-cycle H labelled +1; this may be constructed explicitly via the boundaries of the (−)-clusters intersecting ∂3n . Since H is outermost, the conditional measure of µ1 (given Dn ), restricted to 3, is stochastically smaller than µ0 . On letting n → ∞, we obtain µ1 ( A) ≤ µ0 ( A), which is to say that µ1 ≤st µ0 . By (8.34), µ0 = µ1 . By (8.35), µ1 (N + = 0) = 1, so that θ 1 (κ2 , 2) = 0 as claimed.  Last, but definitely not least, we turn towards SLE and random-cluster/Ising models. Stanislav Smirnov has recently proved the convergence of re-scaled boundaries of large clusters of the critical random-cluster model on L 2 to SLE16/3 . The corresponding critical Ising model has spin-cluster boundaries converging to SLE3 . These results are having major impact on our understanding of the Ising model. This8 section ends with two open problems concerning exponential decay and/or SLE. Each Ising spin-configuration σ ∈ {−1, +1} V on a graph G = (V, E) gives rise to a subgraph G σ = (V, E σ ) of G where (8.36)

E σ = {e = hx, yi ∈ E : σx = σ y }.

If G is planar, the boundary of any connected component of G σ corresponds to a cycle in the dual graph G d , and the union of all such cycles is a (random) even subgraph of G d (see the next section). We shall consider the Ising model on the square and triangular lattices, with inverse-temperature β satisfying 0 ≤ β ≤ βc , where βc is the critical value. By (8.5), e−2βc = 1 − pc (2). √ √ We begin with the square lattice L2 , for which pc (2) = 2/(1 + 2). When β = 0, the model amounts to site percolation with density 21 . Since this percolation process has critical point satisfying pcsite > 12 , each spin-cluster of the β = 0 Ising model is subcritical, and in particular has an exponentially-decaying tail. More ± specifically, write x ←→ y if there exists a path of L2 from x to y with constant spin-value, and let ±

Sx = {y ∈ V : x ←→ y}

be the spin-cluster at x, and S = S0 . By the above, there exists α > 0 such that (8.37)

λ0 (|S| ≥ n + 1) ≤ e−αn ,

n ≥ 1,

where λβ denotes the infinite-volume Ising measure. It is standard (and follows from Theorem 8.17(a)) that there is a unique Gibbs measure for the Ising model when β < βc , and this may be extended to the critical case β = βc (see [101] for example). 8 rewrite

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The exponential decay of (8.37) extends throughout the subcritical phase in the following sense. Yasunari Higuchi [123] has proved that (8.38)

λβ (|S| ≥ n + 1) ≤ e−αn ,

n ≥ 1,

where α = α(β) satisfies α > 0 when β < βc . There is a more recent proof of this (and more) by Rob van den Berg [31, Thm 2.4], using the sharp-threshold theorem, Theorem 4.77. Note that (8.38) implies the weaker (and known) statement that the clusters of the q = 2 random-cluster model on L2 have exponentially-decaying tail.9 Inequality (8.38) fails in an interesting manner when the square lattice is replaced by the triangular lattice T. Since pcsite (T) = 12 , the β = 0 Ising model is critical. In particular, the tail of |S| is of power-type and, by Smirnov’s theorem for percolation, the scaling limit of the spin-cluster boundaries is SLE 6 . Furthermore, the process is, in the following sense, critical for all β ∈ [0, βc ]. Since there is a unique Gibbs state for β < βc , λβ is invariant under the interchange of spin-values −1 ↔ +1. Let Rn be a rhombus of the lattice with side-lengths n and axes parallel to the horizontal and one of the diagonal lattice directions, and let An be the event that Rn is traversed from left to right by a + path (i.e., a path ν satisfying σ y = +1 for all y ∈ ν). It is easily seen that the complement of A n is the event that Rn is crossed from top to bottom by a − path (see Figure 5.13 for an illustration of the analogous case of bond percolation on the square lattice). Therefore, λβ ( An ) = 21 ,

(8.39)

0 ≤ β < βc .

Let Sx be the spin-cluster containing x as before, and define rad(Sx ) = max{|z − x| : z ∈ Sx }, where |y| is the graph-theoretic distance from 0 to y. By (8.39), there exists a vertex x such that λβ (rad(Sx ) ≥ n) ≥ (2n)−1 . By the translation-invariance of λβ , 1 , 0 ≤ β < βc . λβ (rad(S) ≥ n) ≥ 2n In conclusion, the tail of rad(S) is of power-type for all β ∈ [0, βc ).10 It is believed that the SLE6 cluster-boundary limit ‘propagates’ from β = 0 all the way to β < βc . When β = βc , the corresponding limit is the same as that for the square lattice, namely SLE3 , see [62].

9 Mention

Graham-Grimmett Balint-Camia-Meester

10 Mention

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Random even graphs

149

8.6 Random even graphs We call a subset F of the edge-set of G = (V, E) even if each vertex x ∈ V is incident to an even number of elements of F, and we write E for the set of even subsets F. The subgraph (V, F) of G is even if F is even. It is standard that every even set F may be decomposed as an edge-disjoint union of cycles. Let p ∈ [0, 1). The random even subgraph of G with parameter p is that with law (8.40)

η p (F) =

1 |F | p (1 − p)|E \F | , Ze

where Ze =

X

F ∈E

F ∈ E,

p |F | (1 − p)|E \F | .

When p = 21 , we talk of a uniform random even subgraph. We may express η p in the following way. Let φ p = φ p,1 be product measure with density p on  = {0, 1} E . For ω ∈ , let ∂ω denote the set of vertices x ∈ V that are incident to an odd number of ω-open edges. Then η p (F) =

φ p (ω F ) , φ p (∂ω = ∅)

F ∈ E,

where ω F is the edge-configuration whose open set is F. In other words, φ p describes the random subgraph of G obtained by randomly and independently deleting each edge with probability 1 − p, and η p is the law of this random subgraph conditioned on its being even. Let λβ be the Ising measure on a graph H with inverse temperature β ≥ 0, presented in the form   X 1 (8.41) λβ (σ ) = exp β σx σ y , σ = (σx : x ∈ V ) ∈ 6, ZI e=hx,yi∈E

with 6 = {−1, +1}V . See (7.17) and (7.19). A spin configuration σ gives rise to a subgraph G σ = (V, E σ ) of G with E σ given in (8.36) as the set of edges whose endpoints have like spin. When G is planar, the boundary of any connected component of G σ corresponds to a cycle in the dual graph G d , and the union of all such cycles is a (random) even subgraph of G d . A glance at (8.3) informs us that the law of this even graph is ηr where r = e−2β . 1−r

Note that r ≤ 21 . Thus, one way of generating a random even subgraph of a planar graph G = (V, E) with parameter r ∈ [0, 21 ] is to take the dual of the graph G σ with σ is chosen with law (8.41), and with β = β(r ) chosen suitably. c G. R. Grimmett 6 February 2009

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Random-Cluster Model

[8.6]

The above recipe may be cast in terms of the random-cluster model on the planar graph G. First, we sample ω according to the random-cluster measure φ p,q with p = 1−e−2β and q = 2. To each open cluster of ω we allocate a random spin taken uniformly from {−1, +1}. These spins are constant on clusters and independent between clusters. By the discussion of Section 8.1, the resulting spin-configuration σ has law λβ . The boundaries of the spin-clusters may be constructed as follows from ω. Let C 1 , C 2 , . . . , C c be the external boundaries of the open clusters of ω, and let ξ1 , ξ2P , . . . , ξc be independent Bernoulli random variables with parameter 1 . The sum i ξi Ci , with addition interpreted as symmetric difference, has law 2 ηr . It turns out that one may generate a random even subgraph of a graph G from the random-cluster model on G, for an arbitrary, possibly non-planar, graph G. We consider first the uniform case of η p with p = 21 . We identify the family of all spanning subgraphs of G = (V, E) with the family of all subsets of E (the word ‘spanning’ indicates that these subgraphs have the original vertex-set V ). This family can further be identified with  = {0, 1} E = Z2E , and is thus a vector space over Z2 ; the operation + of addition is componentwise addition modulo 2, which translates into taking the symmetric difference of edge-sets: F1 + F2 = F1 4 F2 for F1 , F2 ⊆ E. The family E of even subgraphs of G forms a subspace of the vector space Z 2E , since F1 4 F2 is even if F1 and F2 are even. In particular, the number of even subgraphs of G equals 2c(G) where c(G) = dim(E ). The quantity c(G) is thus the number of independent cycles in G, and, as is well known, (8.42) Cf. (8.26).

c(G) = |E| − |V | + k(G).

(8.43) Theorem [101]. Let C 1 , C 2 , . . . , C c be a maximal set of independent cycles in G. Let ξ1 , ξP 2 , . . . , ξc be independent Bernoulli random variables with 1 parameter 2 . Then i ξi Ci is a uniform random even subgraph of G. P Proof. Since every linear combination i ψi Ci , ψ ∈ {0, 1}c , is even, and since every even graph may be expressed uniquely in this form, the uniform measure on {0, 1}c generates the uniform measure on E . 

One standard way of choosing such a set C 1 , C 2 , . . . , C c , when G is planar, is given as above by the external boundaries of the finite faces. Another is as follows. Let (V, F) be a spanning subforest of G, that is, the union of a spanning tree from each component of G. It is well known, and easy to check, that each edge ei ∈ E \ F can be completed by edges in F to a unique cycle C i . These cycles form a basis of E . By Theorem 8.43, we may therefore find a random uniform subset of the C j by choosing a random uniform subset of E \ F. We show next how to couple the q = 2 random-cluster model and the random even subgraph of G. Let p ∈ [0, 21 ], and let ω be a realization of the randomcluster model on G with parameters 2 p and q = 2. Let R = (V, γ ) be a uniform random even subgraph of (V, η(ω)). c G. R. Grimmett 6 February 2009

[8.6]

Random even graphs

151

(8.44) Theorem [101]. The graph R = (V, γ ) is a random even subgraph of G with parameter p. This recipe for random even subgraphs provides a neat method for their simulation, provided p ≤ 21 . One may sample from the random-cluster measure by the method of coupling from the past (see [178]), and then sample a uniform random even subgraph from the outcome, as above. If G itself is even, we can further e from η1− p and sample from η p for p > 12 by first sampling a subgraph (V, F) e then taking the complement (V, E \ F), which has the distribution η p . One may adapt this argument to obtain an efficient method for sampling from η p for p > 21 and general G (see Exercise 8.16). When G is planar, this amounts to sampling from an antiferromagnetic Ising model on its dual graph. There is a converse to Theorem 8.44. Take a random even subgraph (V, F) of G = (V, E) with parameter p ≤ 21 . To each e ∈ / F, we assign an independent random colour, blue with probability p/(1 − p) and red otherwise. Let B be obtained from F by adding in all blue edges. It is left as an exercise to show that the graph (V, B) has law φ2 p,2 . Proof of Theorem 8.44. Let g ⊆ E be even, and let ω be a sample configuration of the random-cluster model on G. By the above,  −c(ω) 2 if g ⊆ η(ω), P(γ = g | ω) = 0 otherwise, where c(ω) = c(V, η(ω)) is the number of independent cycles in the ω-open subgraph. Therefore, X P(γ = g) = 2−c(ω) φ2 p,2 (ω). ω: g⊆η(ω)

By (8.42),

P(γ = g) ∝

∝

X

(2 p)|η(ω)| (1 − 2 p)|E \η(ω)| 2k(ω)

ω: g⊆η(ω)

X

ω: g⊆η(ω)

p |η(ω)| (1 − 2 p)|E \η(ω)|


1 |η(ω)|−|V |+k(ω) 2

= [ p + (1 − 2 p)]|E \g| p |g|

= p |g| (1 − p)|E \g| ,

The claim follows.

g ⊆ E. 

The above account of even subgraphs would be gravely incomplete without a reminder of the so-called ‘random-current representation’ of the Ising model. This is a representation of the Ising measure in terms of a random field of loops and lines, and it has enabled a rigorous analysis of the Ising model. See [3, 7, 10] and [98, Chap. 9]. The random-current representation is closely related to the study of random even subgraphs. c G. R. Grimmett 6 February 2009

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[8.7]

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8.7 Exercises 8.1. [107] Let φ p,q be a random-cluster measure on a finite graph G = (V, E) with parameters p and q. Prove that n o d 1 φ p,q ( A) = φ p,q (M1 A ) − φ p,q (M)φ p,q ( A) dp p(1 − p)

for any event A, where M = |η(ω)| is the number of open edges of a configuration ω and 1 A is the indicator function of the event A. 8.2. (continuation) Show that φ p,q satisfies the FKG inequality if q ≥ 1, in that φ p,q ( A ∩ B) ≥ φ p,q ( A)φ p,q (B) for increasing events A, B, but does not generally have this property when q < 1. 8.3. Show that the conditional random-cluster measure on G given that the edge e is closed (respectively, open) is that of φ G\e, p,q (respectively, φG.e, p,q ). 8.4. Show that random-cluster measures φ p,q do not generally satisfy the BK inequality if q > 1. That is, find a finite graph G and increasing events A, B such that φ p,q ( A ◦ B) > φ p,q ( A)φ p,q (B). 8.5. (Important research problem, hard if true) Prove that random-cluster measures satisfy the BK inequality if q < 1. 8.6. Let φ p,q be the random-cluster measure on a finite connected graph G = (V, E). Show, in the limit as p, q → 0 in such way that q/ p → 0, that φ p,q converges weakly to the uniform spanning tree measure UST on G. Identify the corresponding limit as p, q → 0 with p = q. Explain the relevance of these limits to the previous question. 8.7. [81] Comparison inequalities. Use the Holley inequality to prove the following ‘comparison inequalities’ for a random-cluster measure φ p,q on a finite graph: φ p0 ,q 0 ≤st φ p,q

if q 0 ≥ q, q 0 ≥ 1, p 0 ≤ p,

φ p0 ,q 0 ≥st φ p,q

if q 0 ≥ q, q 0 ≥ 1,

p p0 ≥ . q 0 (1 − p 0 ) q(1 − p)

8.8. [9] Show that the wired percolation probability θ 1 ( p, q) on Ld equals the limit of the finite-volume probabilities, in that, for q ≥ 1, 1 θ 1 ( p, q) = lim φ3, p,q (0 ↔ ∂3). 3→Zd

8.9. [98, 156] Mixing. A translation τ of Ld induces a translation of  given by τ (ω)(e) = ω(τ −1 (e)). Let A and B be cylinder events of . Show, for q ≥ 1 and b = 0, 1, that φ bp,q ( A ∩ τ n B) → φ bp,q ( A)φ bp,q (B) c G. R. Grimmett 6 February 2009

as n → ∞.

[8.7]

Exercises

153

The following may help when b = 0, with a similar argument when b = 1. a. Assume A is increasing. Let A be defined on the box 3, and let 1 be a larger box with τ n B defined on 1 \ 3. Use positive association to show that 0 n 0 0 n φ1, p,q ( A ∩ τ B) ≥ φ3, p,q ( A)φ1, p,q (τ B).

b. Let 1 ↑ Zd , and then n → ∞ and 3 ↑ Zd , to obtain lim inf φ 0p,q ( A ∩ τ n B) ≥ φ 0p,q ( A)φ 0p,q (B). n→∞

By applying this to the complement B also, deduce that φ 0p,q ( A ∩ τ n B) → φ 0p,q ( A)φ bp,q (B). 8.10. Ergodicity. Deduce from the result of the previous exercise that the φ bp,q are ergodic. 8.11. Use the comparison inequalities to prove that the critical point p c (q) of the random-cluster model on Zd satisfies pc (1) ≤ pc (q) ≤

q pc (1) , 1 + (q − 1) pc (1)

q ≥ 1.

In particular, 0 < pc (q) < 1 if q ≥ 1 and d ≥ 2. 8.12. Let µ be the ‘usual’ coupling of the Potts measure and the random-cluster measure on a finite graph G. Derive the conditional measures of the first component given the second, and of the second given the first. 8.13. Let q ∈ {2, 3, . . . }, and let G = (V, E) be a finite graph. Let W ⊆ V , W and let σ1 , σ2 ∈ {1, 2, . . . , q}W . Starting from the random-cluster measure φ p,q on G with members of W identified as a single point, explain how to couple the associated Potts measures π(· | σ W = σi ), for i = 1, 2, in such a way that: any vertex x not joined to W in the random-cluster configuration has the same spin in each of the two Potts configurations. Let B ⊆ {1, 2, . . . , q}Y where Y ⊆ V \ W . Show that π(B | σW = σ1 ) − π(B | σW = σ2 ) ≤ φ W (W ↔ Y ). p,q

8.14. Ising mixing and ergodicity. Let φ bp,q be a random-cluster measure on Ld with b ∈ {0, 1} and q ∈ {1, 2, . . . }. If b = 0, we assign a uniformly random element of Q = {1, 2, . . . , q} to each open cluster, constant within clusters and independent between. We do similarly if b = 1 with the difference that any infinite cluster receives spin 1. Show that the ensuing spin-measures π b are the infinite-volume Potts measures with free and 1 boundary conditions, respectively. Using the results of the previous exercise, or otherwise, show that π b is mixing, and hence ergodic, if φ bp,q (0 ↔ ∞) = 0. c G. R. Grimmett 6 February 2009

154

Random-Cluster Model

[8.7]

8.15. [93] Show for the random-cluster model on L2 that pc (q) ≥ κq , where √ √ κq = q/(1 + q) is the self-dual point. 8.16. [101] Make a proposal for generating a random even subgraph of the graph G = (V, E) with parameter p satisfying p > 21 . You may find it useful to prove the following first. Let u, v be distinct vertices in the same component of G, and let π be a path from u to v. Let F be the set of even subsets of E, and F u,v the set of subsets F such that deg F (x) is even if and only if x 6= u, v. [Here, deg F (x) is the number of elements of F incident to x.] Then F and F u,v are put in one–one correspondence by F ↔ F 4 π . 8.17. [101] Let (V, F) be a random even subgraph of G = (V, E) with law / F is coloured blue with probability p/(1 − p), η p where p ≤ 21 . Each e ∈ independently of all other edges. Let B be the union of F with the blue edges. Show that (V, B) has law φ2 p.2 .

c G. R. Grimmett 6 February 2009

9 Quantum Ising Model

The quantum Ising model on a finite graph G may be transformed into a continuum random-cluster model on the set obtained by attaching a copy of the real line to each vertex of G. The ensuing representation of the Gibbs operator is susceptible to probabilistic analysis. One application is to an estimate of entanglement in the one-dimensional system.

9.1 The model The quantum Ising model is defined as follows on the finite graph G = (V, E). To each vertex x ∈ V is associated a quantum spin- 12 with local Hilbert space C2 . N The configuration space H for the system is the tensor product 1 H = x∈V C2 . As basis for the copy of C2 labelled by x ∈ V , we take the two eigenvectors, denoted as     1 0 |+ix = , |−ix = , 0 1 of the Pauli matrix

σx(3)

=



1 0

0 −1



at the site x, with corresponding eigenvalues ±1. The other two Pauli matrices with respect to this basis are:     0 1 0 −i (1) (2) σx = , σx = . 1 0 i 0 In the following, |φi denotes a vector and hφ| its adjoint (or conjugate transpose). N Let D be the set of 2|V | basis vectors |ηi for H of the form |ηi = x |±ix . There is a natural one–one correspondence between D and the space 6 = 6 V = 1 The tensor product U

⊗ V of two vector spaces over F is the dual space of the set of bilinear functionals on U × V . Ref? c G. R. Grimmett 6 February 2009

156

[9.2]

Quantum Ising Model

{−1, +1}V . We may speak of members of 6 as basis vectors, and of H as the Hilbert space generated by 6. Let λ, δ ∈ [0, ∞). The Hamiltonian of the quantum Ising model with transverse field is the matrix (or ‘operator’) (9.1)

H = − 21 λ

X

e=hx,yi∈E

σx(3) σ y(3) − δ

X

σx(1) ,

x∈V

Here, λ is the spin-coupling and δ is the transverse-field intensity. The matrix H operates on vectors (elements of H ) through the operation of each σ x on the component of the vector at x. Let β ∈ [0, ∞) be the parameter known as ‘inverse temperature’. The Hamiltonian H generates the matrix e −β H , and we are concerned with the operation of this matrix on elements of H . We normalize e −β H by its trace, that is, we define the so-called ‘density matrix’ (9.2)

ρG (β) =

1 e−β H , Z G (β)

where (9.3)

Z G (β) = tr(e−β H ) =

X

η∈6

hη|e−β H |ηi.

It turns out that the matrix elements of ρG (β) may be expressed in terms of a type of ‘path integral’ with respect to the continuum random-cluster model on V × [0, β] with parameters λ, δ and q = 2. We explain this in the following two sections. The Hamiltonian H has a unique pure ground state |ψ G i defined at zerotemperature (that is, in the limit as β → ∞) as the eigenvector corresponding to the lowest eigenvalue of H .

9.2 Continuum percolation and random-cluster models The finite graph G = (V, E) may be used as a base for a family of probabilistic models that live not on the vertex-set V but on the ‘continuum’ space V × R. The simplest of these models is continuum percolation, see Section 6.6. We consider here a related model called the continuum random-cluster model. Let β ∈ (0, ∞), and let 3 be the ‘box’ 3 = V × [0, β]. In the notation of Section 6.6, let φ 3,λ,δ denote the probability measure associated with the Poisson processes D x , x ∈ V , and Be , e = hx, yi ∈ E. As sample space we take the set 3 comprising all finite sets of cuts and bridges in 3, and we may assume without loss of generality that no cut is the endpoint of any bridge. For ω ∈ 3 , we write B(ω) and D(ω) for c G. R. Grimmett 6 February 2009

[9.2]

Continuum percolation and random-cluster models

157

the sets of bridges and cuts, respectively, of ω. The appropriate σ -field F 3 is that generated by the open sets in the associated Skorohod topology, see [34, 76]. For a given configuration ω ∈ 3 , let k(ω) be the number of its clusters under the connection relation ↔. Let q ∈ (0, ∞), and define the ‘continuum randomcluster’ probability measure φ3,λ,δ,q by 1 k(ω) q dφ3,λ,δ (ω), ω ∈ 3 , Z for an appropriate normalizing constant, or ‘partition function’, Z = Z 3 (λ, δ, q). The continuum random-cluster model may be studied in very much the same way as the random-cluster model on a lattice, see Chapter 8. The space 3 is a partially ordered space with order relation given by: ω1 ≤ ω2 if B(ω1 ) ⊆ B(ω2 ) and D(ω1 ) ⊇ D(ω2 ). A random variable X : 3 → R is called increasing if X (ω) ≤ X (ω 0 ) whenever ω ≤ ω0 . An event A ∈ F3 is called increasing if its indicator function 1 A is increasing. Given two probability measures µ1 , µ2 on a measurable pair (3 , F3 ), we write µ1 ≤st µ2 if µ1 (X ) ≤ µ2 (X ) for all bounded increasing continuous random variables X :  3 → R. The measures φ3,λ,δ,q have certain properties of stochastic ordering as the parameters 3, λ, δ, q vary. The basic theory will be assumed here, and the reader is referred to [37] for further details. In rough terms, the φ3,λ,δ,q inherit the properties of stochastic ordering and positive association enjoyed by their counterparts on discrete graphs. Of particular value in Section 9.5 is the stochastic inequality (9.4)

(9.5)

dφ3,λ,δ,q (ω) =

φ3,λ,δ,q ≤st φ3,λ,δ ,

q ≥ 1.

We note that the thermodynamic limit may be taken in much the same manner as it was for the discrete random-cluster model, whenever q ≥ 1, and for certain boundary conditions τ . Suppose, for example, that V is a finite connected subgraph of the lattice G = Zd , and assign to the box 3 = V × [0, β] a suitable boundary condition. As described in [98] for the discrete case, if the boundary condition τ τ is chosen in such a way that the measures φ3,λ,δ,q are monotonic as V ↑ Zd , then τ τ the weak limit φλ,δ,q,β = lim V ↑Zd φ3,λ,δ,q exists. One may similarly allow the τ τ limit as β → ∞ to obtain a measure φλ,δ,q = limβ→∞ φλ,δ,q,β . Let G = Zd . Restricting ourselves for convenience to the case of free boundary conditions, we define the percolation probability by θ (λ, δ, q) = φλ,δ,q (|C| = ∞),

where C is the cluster at the origin (0, 0), and |C| denotes the aggregate (onedimensional) Lebesgue measure of the time intervals comprising C. The critical point is defined by λc (Zd , q) = sup{λ : θ (λ, 1, q) = 0}.

In the special case d = 1, the random-cluster model has a property of self-duality that leads to the following conjecture. c G. R. Grimmett 6 February 2009

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[9.3]

Quantum Ising Model

(9.6) Conjecture. The continuum random-cluster model on Z × R with clusterweighting factor satisfying q ≥ 1 has critical value λc (Z, q) = q.

It may be proved by standard means that λc (Z, q) ≥ q. See (8.32) and [98, Sect. 6.2] for the corresponding result on the discrete lattice Z2. The cases q = 1, 2 are special. The statement λc (Z, 1) = 1 is part of Theorem 6.19(b). When q = 2, the method of so-called ‘random currents’ may be adapted to the quantum model with several consequences, of which we highlight the fact that λc (Z, 2) = 2; see [38]. The continuum Potts model on V × R is given as follows. Let q ∈ {2, 3, . . . }. To each cluster of the random-cluster model with cluster-weighting factor q is assigned a ‘spin’ from the space 6 = {1, 2, . . . , q}, different clusters receiving independent spins. The outcome is a function σ : V × R → 6, and this is the spin-vector of a ‘continuum q-state Potts model’ with parameters λ and δ. When q = 2, we refer to the model as a continuum Ising model. It is not hard to see that the law P of the above spin model on 3 = V × [0, β] is given by 1 d P(σ ) = eλL(σ ) dφ3,δ (Dσ ), Z where Dσ is the set of (x, s) ∈ V × [0, β] such that σ (x, s−) 6= σ (x, s+), φ3,δ is the law of a family of independent Poisson processes on the time-lines {x}×[0, β], x ∈ V , with intensity δ, and L(σ ) =

X

hx,yi∈E V

Z

β 0

1{σ (x,u)=σ (y,u)} du

is the aggregate Lebesgue measure of those subsets of pairs of adjacent time-lines on which the spins are equal. As usual, Z is an appropriate constant.

9.3 Quantum Ising via random-cluster In this section we describe the relationship between the quantum Ising model on a finite graph G = (V, E) and the continuum random-cluster model on G × [0, β] with q = 2. We shall see that the density matrix ρ G (β) may be expressed in terms of ratios of probabilities given in terms of the random-cluster model. The roots of the following argument lie in the work of Campanino, von Dreyfus, Klein, and Perez, and the reader is referred to [13] for the final form. Similar geometrical transformations can be made for other certain quantum models, see [14, 167]. First, some notation. Let 3 = V × [0, β], and let 3 be the configuration space of the continuum random-cluster model on 3. For given λ, δ, and q = 2, let φG,β denote the corresponding continuum random-cluster measure on  3 (with free boundary conditions). Thus, for economy of notation we suppress reference to λ and δ. c G. R. Grimmett 6 February 2009

[9.3]

159

Quantum Ising via random-cluster

We next introduce a coupling of edge and spin configurations as in Section 8.1. For ω ∈ 3 , let S(ω) denote the space of all functions s : V × [0, β] → {−1, +1} that are constant on the clusters of ω, and let S be the union of the S(ω) over ω ∈ 3 . Given ω, we may pick an element of S(ω) uniformly at random, and we denote this random element as σ . We shall abuse notation by using φ G,β to denote the ensuing probability measure on the coupled space  3 × S. For s ∈ S and W ⊆ V , we write sW,0 (respectively, sW,β ) for the vector (s(x, 0) : x ∈ W ) (respectively, (s(x, β) : x ∈ W )). We abbreviate s V,0 and sV,β to s0 and sβ , respectively. (9.7) Theorem [13]. The elements of the density matrix ρ G (β) satisfy (9.8)

hη0 |ρG (β)|ηi =

φG,β (σ0 = η, σβ = η0 ) , φG,β (σ0 = σβ )

η, η0 ∈ 6.

Proof. We use the notation of Section 9.1. By (9.1) with ν = the identity matrix2 , (9.9)

1 2

P

hx,yi λI

and I

e−β(H +ν) = e−β(U +V ) ,

where U = −δ

X

σx(1) ,

x∈V

V = − 12

X

e=hx,yi∈E

λ(σx(3) σ y(3) − I).

Although these two matrices do not commute, we may use the so-called Lie–Trotter formula (see, for example, [193]) to express e −β(U +V ) in terms of single-site and two-site contributions due to U and V , respectively. By the Lie–Trotter formula, e−(U +V )1t = e−U 1t e−V 1t + O(1t 2 ). We divide the interval [0, β] into N parts each of length 1t = 1/N , so that e−β(U +V ) = lim (e−U 1t e−V 1t )β/1t . 1t→0

Now expand the exponential, neglecting terms of order o(1t), to obtain (9.10) e−β(H +ν) = Y   Y   β/1t  1 3 lim (1 − δ1t)I + δ1t Px (1 − λ1t)I + λ1t Px,y , 1t→0

x

e=hx,yi

1 + I and P 3 = 1 (σ σ where Px1 = σ(x) y + I). x,y 2 x (3) (3)

2 Note that hη 0 |e J +c

|ηi = ec hη0 |e J |ηi, so the introduction of ν into the exponent is harmless.

c G. R. Grimmett 6 February 2009

160

[9.3]

Quantum Ising Model

As noted earlier, we think of members of 6 = {−1, +1} V as basis vectors of H , and of H as the Hilbert space generated by 6. The product (9.10) contains a collection of operators acting on sites x and on neighbouring pairs hx, yi. We partition the time interval [0, β] into N time-segments labelled 1t1, 1t2 , . . . , 1t N , each of length 1t. On neglecting terms of order o(1t), we may see that each given time-segment arising in (9.10) contains exactly one of: the identity matrix 3 . Each such matrix occurs in I; a matrix of the form Px1 ; a matrix of the form Px,y the time-segment with a certain weight. Let us consider the actions of these matrices on the states |ηi for each time interval 1ti , i ∈ {1, 2, . . . , N }. The matrix elements of the single-site operator at x are given by hη0 |σx(1) + I|ηi ≡ 1.

(9.11)

This is easily checked by exhaustion. When this matrix occurs in some timesegment 1ti , we place a mark in the interval {x} × 1ti , and we call this mark a cut. Such a cut has a corresponding weight δ1t + o(1t). The matrix element involving the neighbouring pair hx, yi yields, as above, (9.12)

1 0 (3) (3) 2 hη |σx σ y

+ I|ηi =



1 0

if ηx = η y = η0x = η0y , otherwise.

When this occurs in some time-segment 1ti , we place a bridge between the intervals {x} × 1ti and {y} × 1ti . Such a bridge has a corresponding weight λ1t + o(1t). In the limit 1t → 0, the spin operators generate thus a Poisson process with intensity δ of cuts in each time-line {x} × [0, β], and a Poisson process with intensity λ of bridges between each pair {x} × [0, β], {y} × [0, β] of time-lines, for neighbouring x and y. This is an independent family of Poisson processes. We write Dx for the set of cuts at the site x, and Be for the set of bridges corresponding to an edge e = hx, yi. The configuration space is the set 3 containing all finite sets of cuts and bridges, and we may assume without loss of generality that no cut is the endpoint of any bridge. For two points (x, s), (y, t) ∈ 3, we write (x, s) ↔ (y, t) if there exists a cutfree path from the first to the second that traverses time-lines and bridges. A cluster is a maximal subset C of 3 such that (x, s) ↔ (y, t) for all (x, s), (y, t) ∈ C. Thus the connection relation ↔ generates a continuum percolation process on 3, and we write φ3,λ,δ for the probability measure corresponding to the weight function on the configuration space 3 . That is, φ3,λ,δ is the measure governing a family of independent Poisson processes of cuts (with intensity δ) and of bridges (with intensity λ). The ensuing percolation process has appeared in Section 6.6. Equations (9.11)–(9.12) are to be interpreted in the following way. In calculating the operator e−β(H +ν) , one averages over contributions from realizations of the Poisson processes, on the basis that the quantum spins are constant on every c G. R. Grimmett 6 February 2009

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Quantum Ising via random-cluster

Figure 9.1. An example of a space–time configuration contributing to the Poisson integral (9.18). The cuts are shown as circles and the distinct connected clusters are indicated with different line-types.

cluster of the corresponding percolation process, and each such spin-function is equiprobable. More explicitly, (9.13)

e

−β(H +ν)

=

Z



dφ3,λ,δ (ω) T

Y

Px1 (t)

(x,t)∈D

Y

3 (t 0 ) Px,y

(hx,yi,t 0)∈B



,

where T denotes the time-ordering of the terms in the products, and B (respectively, D) is the set of all bridges (respectively, cuts) of the configuration ω ∈  3 . Let ω ∈ 3 . Let µω be the counting measure on the space S(ω) of functions s : V × [0, β] → {−1, +1} that are constant on the clusters of ω. Let K (ω) be the time-ordered product of operators in (9.13). We may evaluate the matrix elements of K (ω) by inserting the ‘resolution of the identity’ X

(9.14)

η∈6

|ηihη| = I

between any two factors in the product, obtaining thus that (9.15)

hη0 |K (ω)|ηi =

X

s∈S(ω)

c G. R. Grimmett 6 February 2009

1{s0 =η} 1{sβ =η0 } ,

η, η0 ∈ 6.

162

[9.3]

Quantum Ising Model

This is the number of spin-allocations to the clusters of ω with given spin-vectors at times 0 and β. The matrix elements of ρG (β) are therefore given by Z 1 0 1{s0 =η} 1{sβ =η0 } dµω (s) dφ3,λ,δ (ω), (9.16) hη |ρG (β)|ηi = Z G,β for η, η0 ∈ 6, where (9.17)

Z G,β = tr(e−β(H +ν) ).

For η, η0 ∈ 6, let Iη,η0 be the indicator function of the event (in 3 ) that, for all x, y ∈ V , if (x, 0) ↔ (y, 0), then η x = η y ,

if (x, β) ↔ (y, β), then η0x = η0y , if (x, 0) ↔ (y, β), then ηx = η0y .

This is the event that the pair (η, η 0 ) of initial and final spin-vectors is ‘compatible’ with the random-cluster configuration. We have that Z X 1 0 (9.18) hη |ρG (β)|ηi = dφ3,λ,δ (ω) 1{s0 =η} 1{sβ =η0 } Z G,β s∈6(ω) Z
k(ω) 1 dφ3,λ,δ (ω) = 2k(ω)+k(ω) Iη,η0 12 Z G,β 1 = φG,β (σ0 = η, σβ = η0 ). η, η0 ∈ 6, Z G,β where k(ω) is the number of clusters of ω containing no point of the form (v, 0) or (v, β), for v ∈ V , and k(ω) = k(ω) − k(ω) is the number remaining. See Figure 9.1 for an illustration of the space–time configurations contributing to the Poisson integral (9.18). On setting η = η0 in (9.18) and summing over η ∈ 6, we find that (9.19)

Z G,β = φG,β (σ0 = σβ ),

as required.



This section closes with an alternative expression for the formula of Theorem 9.7. We consider ‘periodic’ boundary conditions on 3 obtained by, for each x ∈ V , identifying (x, 0) and (x, β). Let k per (ω) be the number of open clusters per of ω with periodic boundary conditions, and φ G,β be the corresponding randomcluster measure. By (9.18), Z 1 2k(ω) Iη,η0 dφ3,λ,δ (ω). hη0 |ρG (β)|ηi = Z G,β c G. R. Grimmett 6 February 2009

[9.4]

Long-range order

By setting η0 = η and summing, (9.20)

1=

X

η∈6

1

hη|ρG (β)|ηi =

Z G,β

Z

2k(ω) 2k

per

(ω)−k(ω)

163

dφ3,λ,δ (ω),

whence Z G,β equals the normalizing constant for the periodic random-cluster per measure φG,β .

9.4 Long-range order The two-point connectivity function
per τG,β (x, y) = φG,β (x, 0) ↔ (y, 0) ,

x, y ∈ V,

for the periodic random-cluster measure turns out to be a natural measure of longrange order in the quantum Ising model, with the order parameter of the latter given as in the next theorem. (9.21) Theorem [13]. We have that
τG,β (x, y) = tr ρG (β)σx(3) σ y(3) ,

x, y ∈ V.

Proof. The argument leading to (9.18) is easily adapted to obtain tr ρG (β) · Now,

(3) (3) 1 2 (σx σ y

X

η: ηx =η y




+ I) =

Iη,η =



2k

per

1 Z G,β

Z

2

k(ω)

(ω)−k(ω)

per 2k (ω)−k(ω)−1

 X

η: ηx =η y



Iη,η dφ3,λ,δ (ω).

if (x, 0) ↔ (y, 0), if (x, 0) ↔ / (y, 0),

whence, by the remark at the end of the last section,
tr ρG (β) · 21 (σx(3) σ y(3) + I) = τG,β (x, y) + 21 (1 − τG,β (x, y)),

and the claim follows.



Expand discussion of critical point? The infinite-volume limits of the quantum Ising model on G are obtained in the ‘ground state’ as β → ∞, and in the spatial limit as |V | → ∞. The paraphernalia of the discrete random-cluster model may be adapted to the current continuous setting in order to understand the issues of existence and uniqueness of these limits. c G. R. Grimmett 6 February 2009

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[9.5]

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We do not investigate that here. Instead, we point out that the behaviour of the twopoint connectivity function, after taking the limits β → ∞, |V | → ∞, depends pivotally on the existence or not of an unbounded cluster in the random-cluster model. Let φλ,δ be the infinite-volume measure, and let θ (λ, δ) = φλ,δ (C 0 is unbounded)

be the percolation probability. Then τλ,δ (x, y) → 0 as |x − y| → ∞, when θ (λ, δ) = 0. On the other hand, by the FKG inequality and the (a.s.) uniqueness of the unbounded cluster, τλ,δ (x, y) ≥ θ (λ, δ)2 ,

implying that τλ,δ (x, y) is bounded uniformly away from 0 when θ (λ, δ) > 0. A more detailed investigation of the infinite-volume limits and their implications for the quantum Ising model may be found in [13]. As pointed out there, the situation is more interesting in the ‘disordered’ setting, when the λ e and δx are themselves random variables.

9.5 Entanglement in one dimension It is shown next how the random-cluster analysis of the last section enables progress with the problem of so-called entanglement in one dimension. The principle reference for the work of this section is [106]. Let G = (V, E) be a finite graph, and let W ⊆ V . A considerable effort has been spent on understanding the so-called ‘entanglement’ of the spins in W relative to those of V \ W , in the (ground state) limit as β → ∞. This is already a hard problem when G is a finite subgraph of the line Z. Various methods have been used in this case, and a variety of results, some rigorous, obtained. The first step in the definition of entanglement is to define the reduced density matrix ρGW (β) = trV \W (ρG (β)), N where the trace is taken over the Hilbert space H V \W = x∈V \W C2 of spins of vertices of V \ W . An analysis (omitted here) exactly parallel to that leading to Theorem 9.7 allows the following representation of the matrix elements of ρ GW (β). (9.22) Theorem [106]. The elements of the reduced density matrix ρ GW (β) satisfy (9.23)

hη0 |ρGW (β)|ηi =

φG,β (σW,0 = η, σW,β = η0 | E) , φG,β (σ0 = σβ | E)

where E is the event that σ V \W,0 = σV \W,β .

η, η0 ∈ 6W ,

N Let DW be the set of 2|W | vectors |ηi of the form |ηi = x∈W |±ix , and write HW for the Hilbert space generated by D W . Just as before, there is a natural c G. R. Grimmett 6 February 2009

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165

one–one correspondence between D W and the space 6W = {−1, +1}W , and we shall thus regard HW as the Hilbert space generated by 6 W . We may write ρG = lim ρG (β) = |ψG ihψG | β→∞

for the density matrix corresponding to the ground state of the system, and similarly (9.24)

ρGW = trV \W (|ψG ihψG |) = lim ρGW (β). β→∞

The entanglement of the spins in W may be defined as follows. (9.25) Definition. The entanglement of the vertex-set W relative to its complement V \ W is the entropy (9.26)

SGW = −tr(ρGW log2 ρGW ).

The behaviour of SGW , for general G and W , is not understood at present. We specialize here to the case of a finite subset of the one-dimensional lattice Z. Let m, L ≥ 0 and take V = [−m, m + L] and W = [0, L], viewed as subsets of Z. We obtain G from V by adding edges between each pair x, y ∈ V with |x − y| = 1. We write ρm (β) for ρG (β), and SmL (respectively, ρmL ) for SGW (respectively, ρGW ). A key step in the study of SmL for large m is a bound on the norm of the difference ρmL − ρnL . The operator norm of a Hermitian matrix3 A is given by k Ak = sup hψ| A|ψi , kψk=1

where the supremum is over all vectors ψ with L 2 -norm 1.

(9.27) Theorem [106]. Let λ, δ ∈ (0, ∞) and write θ = λ/δ. There exist constants C, α, γ depending on θ and satisfying γ > 0 when θ < 1 such that: (9.28)

 kρmL − ρnL k ≤ min 2, C L α e−γ m ,

2 ≤ m ≤ n < ∞, L ≥ 1.

One would expect that γ may be taken in such a manner that γ > 0 under the weaker assumption λ/δ < 2, but this has not yet quite been proved (cf. Conjecture 9.6). The constant γ is, apart from a constant factor, the reciprocal of the correlation length of the associated random-cluster model. Proved, no? Inequality (9.28) is proved by the following route. Consider the continuum random-cluster model with q = 2 on the space–time graph 3 = V × [0, β] with ‘partial periodic top/bottom boundary conditions’; that is, for each x ∈ V \ W , 3A

matrix is called Hermitian if it equals its conjugate transpose.

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[9.5]

Quantum Ising Model p

we identify the two vertices (x, 0) and (x, β). Let φm,β denote the associated random-cluster measure on 3 . To each cluster of ω ∈ 3 we assign a random p spin from {−1, +1} in the usual manner, and we abuse notation by using φ m,β to denote the measure governing both the random-cluster configuration and the spin configuration. Let p am,β = φm,β (σW,0 = σW,β ), noting that

am,β = φm,β (σ0 = σβ | E)

as in (9.23). By Theorem 9.22, (9.29)

p

hψ|ρmL (β) − ρnL (β)|ψi

=

φm,β (c(σW,0 )c(σW,β )) am,β

where c : {−1, +1}W → C and ψ=

X

η∈6W

p

−

φn,β (c(σW,0 )c(σW,β) ) an,β

,

c(η)η ∈ HW .

The random-cluster property of ratio weak-mixing is used in the derivation of (9.28) from (9.29). This may be stated roughly as follows. Let A and B be events in the continuum random-cluster model that are defined on regions R A and R B of space, respectively. What can be said about the difference φ( A ∩ B) − φ( A)φ(B) when the distance d(R A , R B ) between R A and R B is large? It is not hard to show that this difference is exponentially small in the distance, so long as the randomcluster model has exponentially-decaying connectivities, and such a property is called ‘weak mixing’. It is harder to show a similar bound for the difference φ( A | B) − φ( A), and such a bound is termed ‘ratio weak-mixing’. The ratio weak-mixing property of random-cluster model has been investigated in [18, 19] for the discrete case and in [106] for the continuum model. At the final step of the proof of Theorem 9.27, the random-cluster model is compared via (9.5) with the continuum percolation model of Section 6.6, and the exponential decay of Theorem 9.27 follows by Theorem 6.19. A logarithmic bound on the entanglement entropy follows for sufficiently small λ/δ. (9.30) Theorem [106]. Let λ, δ ∈ (0, ∞) and write θ = λ/δ. There exists θ0 ∈ (0, ∞) such that: for θ < θ0 , there exists K = K (θ ) < ∞ such that SmL ≤ K log2 L ,

m ≥ 0, L ≥ 2.

Discuss: spectra are close, so... A stronger result is expected, namely that the entanglement SmL is bounded above, uniformly in L, whenever θ is sufficiently small, and perhaps for all θ < θ c c G. R. Grimmett 6 February 2009

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Exercises

167

where θc = 2 is the critical point. It is not clear whether this is provable by the methods of this chapter. See Conjecture 9.6 above, and the references in [106]. There is no rigorous picture known of the behaviour of SmL for large θ , or of the corresponding quantity in dimensions d ≥ 2, although Theorem 9.27 has a counterpart in this setting. Theorem 9.30 may be extended to the disordered system in which the intensities λ, δ are independent random variables indexed by the vertices and edges of the underlying graph, subject to certain conditions on these variables (cf. Theorem 6.20 and the preceding discussion).

9.6 Exercises 9.1. Explain in what manner the continuum random-cluster measure φ λ,1,q on

Z × R is ‘self-dual’ when λ = q and q ≥ 1.

c G. R. Grimmett 6 February 2009

10 Interacting Particle Systems

The contact, voter, and exclusion models are examples of so-called interacting particle systems. Each is a Markov process in continuous time, with state space {0, 1} V for some countable set V . In the voter model, each element of V may be in either of two states, and its state flips at a rate that is a weighted average of the states of the other elements. When V = Z d , the analysis of the voter model hinges on the recurrence or transience of an associated Markov chain. When d = 2 and the model is generated by simple random walk, the only invariant measures are the two point masses on the (two) states representing unanimity. The picture is more complicated when d ≥ 2, owing to the transience of the random walk. In the exclusion model, a set of particles moves about V according to a ‘symmetric’ Markov chain, subject to exclusion. We shall assume that V = Zd , and that the Markov chain is translation-invariant. It turns out that the product measures are invariant for this process, and furthermore that these are exactly the extremal invariant measures.

10.1 Introductory remarks There are many beautiful problems of physical type that may be modelled as Markov processes on the compact state space 6 = {0, 1} V for some countable set V . Amongst the most studied to date by probabilists are the contact, voter, and exclusion models1 . This significant branch of modern probability theory had its nascence around 1970 in the work of Roland Dobrushin, Frank Spitzer, and others, and has been brought to maturity through the work of Thomas Liggett and colleagues. The basic references are Liggett’s two volumes [148, 150], see also [151]. The general theory of Markov processes, with its intrinsic complexities, is avoided here. The three processes of this chapter may be constructed via ‘graphical representations’ involving independent random walks. There is a general approach 1 We

say nothing about the stochastic Ising model here.

c G. R. Grimmett 6 February 2009

[10.1]

Introductory remarks

169

to such important matters as existence, for an account of which the reader is referred to [148]. The two observations of note are that the state space 6 is compact, and that the Markov processes (ηt : t ≥ 0) of this section are Feller processes, which is to say that the transition measures are weakly continuous functions of the initial state2 . For a given Markov process, the two main questions are to identify the set of invariant measures, and to identify the ‘basin of attraction’ of a given invariant measure. The processes of this chapter will always possess a non-empty set I of invariant measures, although it is not always possible to describe all members of this set explicitly. Since I is a convex set of measures, it suffices to describe its extremal elements. We shall see that, in certain circumstances, |I| = 1, and this may be interpreted as the absence of long-range order. Since V is infinite, 6 is uncountable. We normally specify the transition operators of a Markov chain on such 6 by specifying its generator. This is an operator G acting on an appropriate dense subset of C(6), the space of continuous functions on 6 endowed with the product topology and the supremum norm. It is determined by its values on the space C(6) of cylinder functions, being the set of functions that depend on only finitely many coordinates in 6. For f ∈ C(6), we write G f in the form (10.1)

G f (η) =

X

η0 ∈6

c(η, η0 )[ f (η0 ) − f (η)],

η ∈ 6,

for some function c. For η 6= η 0 , we think of c(η, η0 ) as being the rate at which the process, when in state η, jumps to state η 0 . The processes ηt possesses a transition semigroup (St : t ≥ 0) acting on C(6) and given by (10.2)

St f (η) = E η ( f (ηt )),

η ∈ 6,

where E η denotes expectation under the assumption η0 = η. Under certain conditions on the process, the transition semigroup is related to the generator by the formula (10.3)

St = exp(t G),

suitably interpreted according to the Hille–Yosida theorem, see [148, Sect. I.2]. The semigroup acts on probability measures by (10.4)

µSt ( A) =

Z

6

P η (ηt ∈ A) dµ(η).

2 Let C(6) denote the space of continuous functions on 6 endowed with the product topology and the supremum norm. The process ηt is Feller if: for f ∈ C(6), f t (η) = E η ( f (ηt )) defines a function belonging to C(6). Here, E η denotes expectation with initial state η.

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Interacting Particle Systems

[10.2]

A probability measure µ on 6 is called invariant for the process η t if µSt = µ for all t. Under suitable conditions, µ is invariant if and only if Z (10.5) G f dµ = 0 for all f ∈ C(6).

In the remainder of this chapter we shall encounter certain constructions of Markov processes on 6, and all such constructions will satisfy the conditions alluded to above.

10.2 Contact model Let G = (V, E) be a connected graph with bounded vertex-degrees. The state space is 6 = {0, 1}V , where the local state 1 (respectively, 0) represents ‘ill’ (respectively, ‘healthy’). Ill vertices recover at rate δ, and healthy vertices become ill at a rate that is linear in the number of ill neighbours. See Chapter 6. One proceeds more formally as follows. For η ∈ 6 and x ∈ V , let η x denote the state obtained from η by flipping the local state of x. That is,  1 − η(x) if y = x, ηx (y) = η(y) otherwise. We let the function c of (10.1) be given by  δ if η(x) = 1, c(η, ηx ) = λ|{y ∼ x : η(y) = 1}| if η(x) = 0,

where λ and δ are strictly positive constants. If η 0 = ηx for no x ∈ V , and η0 6= η, we set c(η, η0 ) = 0. We saw in Chapter 6 that the point mass on the empty set, ν = δ∅ , is the minimal invariant measure of the process, and that there exists a maximal invariant measure ν obtained as the weak limit of the process with initial state V . As remarked at the end of Section 6.3, when G = Ld , the set of extremal invariant measures is exactly Ie = {δ∅ , ν}, and δ∅ = ν if and only if there is no percolation in the associated oriented percolation model in continuous time. Of especial use in proving these facts was the coupling of contact models in terms of Poisson processes of cuts and (directed) bridges. We revisit duality briefly, see Theorem 6.1. For η ∈ 6 and A ⊆ V , let  Y 1 if η(x) = 0 for all x ∈ A, (10.6) H (η, A) = [1 − η(x)] = 0 otherwise. x∈A The conclusion of Theorem 6.1 may be expressed more generally as: E A (H ( At , B)) = E B (H ( A, Bt )),

where At (respectively, Bt ) denotes the contact model with initial state A 0 = A (respectively, B0 = B). This may seem a strange way to express the duality relation, but its relevance may become clearer soon. c G. R. Grimmett 6 February 2009

[10.3]

Voter model

171

10.3 Voter model Let V be a countable set, and let P = ( p x,y : x, y ∈ V ) be the transition matrix of a Markov chain on V . The associated voter model is given by choosing X (10.7) c(η, ηx ) = px,y y: η(y)6=η(x)

in (10.1). The meaning of this is as follows. Each member of V is an individual in a population, and may have either of two opinions at any given time. Let x ∈ V . At times of a rate-1 Poisson process, x selects a random y according to the measure px,y , and adopts the opinion of y. It turns out that the behaviour of this model is closely related to the transience/recurrence of the chain with transition matrix matrix P, and of properties of its harmonic functions. The voter model has two absorbing states, namely all 0 and all 1, and we denote by δ0 and δ1 the point masses on these states. Any convex combination of δ 0 and δ1 is invariant also, and thus one asks for conditions under which every invariant measure is of this form. A duality relation will enable us to answer this question. It is helpful to draw the graphical representation of the process. With each x ∈ V is associated a ‘time-line’ [0, ∞), and on each such time-line is marked the set of epochs of a Poisson process Po x with intensity 1. Different time-lines possess independent Poisson processes. Associated with each epoch of the Poisson process at x is a vertex y chosen at random according to the transition matrix P. The meaning of y is as above. Consider the state of vertex x at time t. We imagine a particle that is at position x at time t, and we write X x (0) = x. When we follow the time-line x × [0, t] backwards in time, that is, from (x, t) to (x, 0), we encounter a first point (first in this reversed ordering of time) in Pox . At this time the particle jumps to the selected neighbour of x. Continuing likewise, the particle performs a simple random walk about V . Writing X x (t) for its position at time 0, the (voter) state of x at time t is precisely that of X x (t) at time 0. Suppose we proceed likewise starting from two vertices x and y at time t. Tracing the states of x and y backwards, each follows a Markov chain with transition matrix P, denoted X x and X y respectively. These chains are independent until the first time (if ever) at which they meet. When they meet, they ‘coalesce’: if they ever occupy the same vertex at any given time, then they follow the same trajectory subsequently. We state this as follows. The presentation here is somewhat informal, and may be made more complete as in [148]. We write (ηt : t ≥ 0) for the voter process, and S for the set of finite subsets of V . (10.8) Theorem. Let A ∈ S, η ∈ 6, and let ( A t : t ≥ 0) be a system of coalescing random walks beginning on the set A 0 = A. Then, Pη (ηt ≡ 1 on A) = P A (η ≡ 1 on At ), c G. R. Grimmett 6 February 2009

t ≥ 0.

172

[10.3]

Interacting Particle Systems

This may be expressed in the form with

Eη (H (ηt , A)) = E A (H (η, At )),

H (η, A) =

Y

η(x).

x∈A

Proof. Each side of the equation is the measure of the complement of the event that, in the graphical representation, there is a path from (x, 0) to (a, t) for some x with η(x) = 0 and some a ∈ A. 

For simplicity, we restrict ourselves henceforth to a case of special interest, namely with V the vertex-set Zd of the d-dimensional lattice with d ≥ 1, and with px,y = p(x − y) for some function p. In the special case of simple random walk, where 1 (10.9) p(z) = , |z| = 1, 2d we have that η(x) flips at a rate equal to the proportion of neighbours of x whose states disagree with the current value η(x). The case of general P is treated in [148]. Let X t and Yt be independent random walks on Zd with rate-1 exponential holding times, and jump distribution p x,y = p(y − x). The difference X t − Yt is a Markov chain also. If X t − Yt is recurrent, we say that we are in the recurrent case, otherwise the transient case. The analysis of the voter model is fairly simple in the recurrent case. (10.10) Theorem. Assume we are in the recurrent case. 1. Ie = {δ0 , δ1 }. 2. If µ is a probability measure on 6 with µ(η(x) = 1) = α for all x ∈ Zd , then µSt ⇒ (1 − α)δ0 + αδ1 as t → ∞.

The situation is quite different in the transient case. We may construct a family of distinct invariant measures να indexed by α ∈ [0, 1], and we do this as follows. Let φα be product measure on 6 with density α. We shall show the existence of the weak limits να = limt→∞ φα St , and it turns out that the να are exactly the extremal invariant measures. A partial proof of the next theorem is provided below. (10.11) Theorem. Assume we are in the transient case. 1. The weak limits να = limt→∞ φα St exist. 2. The να are translation-invariant and ergodic3 , with density να (η(x) = 1) = α.

3 A probability measure µ on 6 is ergodic if any shift-invariant event has µ-probability either 0 or 1. It is standard that the ergodic measures are extremal within the class of translation-invariant measures, see [86] for example.

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Voter model

173

3. Ie = {να : α ∈ [0, 1]}.

We return briefly to the voter model corresponding to simple random walk on

Zd , see (10.9). It is an elementary consequence of Po´ lya’s theorem, Theorem

1.32, that we are in the recurrent case if and only d ≤ 2.

Proof of Theorem 10.10. By assumption, we are in the recurrent case. Let x, y ∈ Zd . By duality, Theorem 10.8, and recurrence,
(10.12) P(ηt (x) 6= ηt (y)) ≤ P X x (u) 6= X y (u) for 0 ≤ u ≤ t →0

For A ∈ S, A 6= ∅, and, by (10.12), (10.13)

as t → ∞.

P(ηt is non-constant on A) ≤ P A (| At | > 1),

P A (| At | > 1) ≤

X

x,y∈A

→0

P X x (u) 6= X y (u) for 0 ≤ u ≤ t

as t → ∞.




It follows that, for any extremal invariant measure µ, the µ-measure of the set of constant configurations is 1. Only the convex combinations of δ 0 and δ1 have this property. Let µ be a probability measure with density α, as in the statement of the theorem, and let A ∈ S, A 6= ∅. By Theorem 10.8 again, Z µSt ({η : η ≡ 1 on A}) = Pη (ηt ≡ 1 on A) µ(dη) Z = P A (η ≡ 1 on At ) µ(dη) Z = P A (η ≡ 1 on At , | At | > 1) µ(dη) X + P A ( At = {y})µ(η(y) = 1), y∈Zd

whence

µSt ({η : η ≡ 1 on A}) − α ≤ 2P A (| At | > 1).

By (10.13), µSt ⇒ (1 − α)δ0 + αδ1 as claimed.

Partial proof of Theorem 10.11. For A ∈ S, A 6= ∅, Z (10.14) φα St (ηt ≡ 1 on A) = Pη (ηt ≡ 1 on A) φα (dη) Z = P A (η ≡ 1 on At ) φα (dη) = E A (α |At | ).

c G. R. Grimmett 6 February 2009



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Interacting Particle Systems

[10.4]

The last expectation converges as t → ∞, since | A t | is non-increasing in t. Using the inclusion–exclusion principle, we deduce that the µSt -measure of any cylinder event has a limit, and therefore the weak limit να exists (see the discussion of weak convergence in Section 2.3). Since the initial state φα is translation-invariant, so is να . We omit the proof of ergodicity, which may be found in [148, 151]. By (10.14) with A = {x}, φα St (η(x) = 1) = α for all t. It may be shown that I is exactly the convex hull of the set {να : α ∈ [0, 1]}. Since the να are ergodic, they are extremal within the class of translation-invariant measures, and therefore (10.15)

Ie ⊇ {να : α ∈ [0, 1]}.

Conversely, take ν ∈ Ie . By the remark above, ν is a mixture of the να . Since ν is extremal, it equals one of the να , whence equality holds in (10.15). The proof of the main statement above is omitted, and may be found in [148, 151]. 

10.4 Exclusion model In this model for a lattice gas, particles jump around the countable set V , subject to the excluded-volume constraint that no more than one particle may occupy any given vertex at any given time. The state space is 6 = {0, 1} V , where the local state 1 represents occupancy by a particle. The dynamics are assumed to proceed as follows. Let P = ( px,y : x, y ∈ V ) be the transition matrix of a Markov chain on V . In order to guarantee the existence of the corresponding exclusion process, we shall assume that X sup px,y < ∞. y∈V x∈V

If the current state is η ∈ 6, and η(x) = 1, the particle at x waits an exponentially distributed time, parameter 1, before it attempts to jump. At the end of this holding time, it chooses a vertex y according to the probabilities p x,y . If, at this instant, y is empty then this particle jumps to y. If y is occupied, the jump is suppressed, and the particle remains at x. Particles are deemed to be indistinguishable. The generator G of the Markov process is given by X px,y [ f (ηx,y ) − f (η)], G f (η) = x,y∈V : η(x)=1, η(y)=0

for cylinder functions f , where η x,y is the state obtained from η by interchanging the local states of x and y, that is,   η(x) if z = y, ηx,y (z) = η(y) if z = x,  η(z) otherwise. c G. R. Grimmett 6 February 2009

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Exclusion model

175

We may construct the process via a graphical representation, as in Section 10.3. For each x ∈ V , we let Pox be a Poisson process with rate 1; these are the times at which a particle at x (if, indeed, x is occupied at the relevant time) attempts to move away from x. With each ‘time’ T ∈ Po x , we associate a vertex Y chosen according to the mass function p x,y , y ∈ V . If x is occupied by a particle at time T , this particle attempts to jump at this instant of time to the new position Y . The jump is successful if Y is empty at time T , otherwise the move is suppressed. It is immediate that the two Dirac measures δ0 and δ1 are invariant. We shall see below that the family of invariant measures is generally much richer than this. The theory is substantially simpler in the symmetric case, and thus we assume henceforth that (10.16)

px,y = p y,x ,

x, y ∈ V.

See [148, Chap. VIII] and [151] for bibliographies for the asymmetric case. If V is the vertex-set of a graph G = (V, E), and P is the transition matrix of simple random walk on G, then (10.16) amounts to the assumption that G be regular. Mention TASEP etc. Duality plays once again a central role in the analysis of the symmetric process. We shall see that the model is self-dual, in the sense of the following Theorem 10.17. Note first that the graphical representation of a symmetric model may be expressed in a slightly simplified manner. For each unordered pair x, y ∈ V , let Pox,y be a Poisson process with intensity p x,y [= p y,x ]. For each T ∈ Pox,y , we interchange the states of x and y at time T . That is, any particle at x moves to y, and vice versa. It is easily seen that the corresponding particle system is the exclusion model. For every x ∈ V , a particle at x at time 0 would pursue a trajectory through V that is determined by the graphical representation, and we denote this trajectory by R x (t), t ≥ 0, noting that R x (0) = x. The processes Rx (·), x ∈ V , are of course dependent. The family (Rx (·) : x ∈ V ) is time-reversible in the following sense. Let t > 0 be given. For each y ∈ V , one may trace the trajectory arriving at (y, t) backwards in time, and we denote the resulting path by B y,t (v), 0 ≤ v ≤ t, with B y,t (0) = y. It is clear by the properties of a Poisson process that the families (Rx (u) : u ∈ [0, t], x ∈ V ) and (B y,t (v) : v ∈ [0, t], y ∈ V ) have the same laws. Let (ηt : t ≥ 0) denote the exclusion model. We distinguish the general model from one that possesses only finitely many particles. Let S be the set of finite subsets of V , and write ( A t : t ≥ 0) for an exclusion process with initial state A0 ∈ S. We think of ηt as a random 0/1-vector, and of A t as a random subset of the vertex-set V . (10.17) Theorem. Consider a symmetric exclusion model on V . For every η ∈ 6 and A ∈ S, (10.18)

Pη (ηt ≡ 1 on A) = P A (η ≡ 1 on At ),

c G. R. Grimmett 6 February 2009

t ≥ 0.

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Proof. The left side of (10.18) equals the probability that, in the graphical representation: for every y ∈ A there exists x ∈ V with η(x) = 1 such that R x (t) = y. By the remarks above, this equals the probability that η(R y (t)) = 1 for every y ∈ A.  (10.19) Corollary. Consider a symmetric exclusion model on V . For each α ∈ [0, 1], the product measure φα on 6 is invariant. Refs here and elsewhere. Proof. Let η be sampled from 6 according to the product measure φ α . We have that P A (η ≡ 1 on At ) = α |At | = α |A| .

By Theorem 10.17, if η0 has law φα , then so does ηt for all t. That is, φα is an invariant measure.  The question thus arises of determining the circumstances under which the set of invariant extremal measures is exactly the set of product measures. Assume for simplicity that: V = Zd , that the transition probabilities are symmetric and translation-invariant in that px,y = p y,x = p(y − x),

x, y ∈ Zd ,

for some function p, and that the associated Markov chain is irreducible. It can be shown in this case (see [148, 151]) that Ie = {φα : α ∈ [0, 1]}, and that µSt ⇒ φα

as t → ∞,

for any translation-invariant and spatially ergodic probability measure µ with α = µ(η(0) = 1). In the more general symmetric non-translation-invariant case on an arbitrary countable set V , the constants α are replaced by the set H of functions α : V → [0, 1] satisfying X px,y α(y), x ∈ V, (10.20) α(x) = y∈V

that is, the bounded harmonic functions, re-scaled if necessary to take values in [0, 1]. [Recall that an irreducible symmetric translation-invariant Markov chain on Zd has only constant bounded harmonic functions4 .] Let µα be the product measure on 6 with µα (η(x) = 1) = α(x). It turns out that the weak limit να = lim µα St t→∞

4 Exercise: Prove this statement. It is an easy consequence of the optional stopping theorem for bounded martingales, whenever the chain is recurrent. See [148, pp. 67–70] for a discussion of the general case.

c G. R. Grimmett 6 February 2009

[10.5]

Exercises

177

exists, and that Ie = {να : α ∈ H }. It may be shown that: να is a product measure if and only if α is a constant function. See [148, 151]. One may find examples for which the set H is large. Let P = ( p x,y ) be the transition matrix of simple random walk on a binary tree T (each of whose vertices has degree 3, see Figure 6.2). Let 0 be a given vertex of the tree, and think of 0 as the root of three disjoint sub-trees of T . Any solution (an : n ≥ 0) to the difference equation (10.21)

2an+1 − 3an + an−1 = 0,

n ≥ 1,

defines a harmonic function α on a given such sub-tree, by α(x) = a n where n is the distance between 0 and x. The general solution to (10.21) is an = A + B( 12 )n ,

where A and B are arbitrary constants. The three pairs ( A, B), corresponding to the three sub-trees at 0, may be chosen in an arbitrary manner, subject to the condition that a0 = A + B is constant between sub-trees. Furthermore, the composite harmonic function on T takes values in [0, 1] if and only if each pair ( A, B) satisfies A, A + B ∈ [0, 1]. There is thus a continuum of admissible non-constant solutions to (10.20), and therefore a continuum of corresponding extremal invariant measures of the associated exclusion model.

10.5 Exercises 10.1. [212] Biased voter model. Each point of the square lattice is occupied, at each time t, by either a benign or a malignant cell. Benign cells invade their neighbours, each neighbour being invaded at rate β, and similarly malignant cells invade their neighbours at rate µ. Suppose there is exactly one malignant cell at time 0, and let κ = µ/β ≥ 1. Show that the malignant cells die out with probability κ −1 . What happens on Zd with d ≥ 2? 10.2. Stochastic Ising model. Let 6 = {−1, +1} V be the state space of a Markov process on the finite graph G = (V, E) in which the state at x ∈ V changes value at rate c(x, σ ) when the state overall is σ . Show that the flip rates  X  c(x, σ ) = exp − σx σ y , y∈∂ x

c0 (x, σ ) =

1 + exp

1 P

y∈∂ x

2σx σ y


,

give rise to time-reversible dynamics with respect to the Ising measure with the same value of β and zero external-field. c G. R. Grimmett 6 February 2009

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Interacting Particle Systems

[10.5]

10.3. A probability measure µ on {0, 1}Z is called exchangeable if the quantity µ({η : η ≡ 1 on A}), as A ranges over the set of finite subsets of Z, depends only on the cardinality of A. Show that every exchangeable µ is invariant for a symmetric exclusion model on Z.

c G. R. Grimmett 6 February 2009

11 Random Graphs

In the Erd˝os–R´enyi random graph G n, p , each pair of vertices is connected by an edge with probability p. We describe the emergence of the giant component when pn ≈ 1, and identify the density of this component as the survival probability of a Poisson branching process. The Hoeffding inequality may be used to show that, for constant p, the chromatic number of G n, p is asymptotic to 12 n/ logπ n where π = 1/(1 − p).

11.1 Erd˝os–R´enyi graphs Let V = {1, 2, . . . , n}, and let (X i, j : 1 ≤ i < j ≤ n) be independent Bernoulli random variables with parameter p. For each pair i < j we place an undirected edge hi, j i between vertices i and j if and only if X i, j = 1. The resulting random graph is named after Erd˝os and R´enyi [75]1 , and it is commonly denoted G n, p . The density p of edges may vary with n, for example, p = λ/n where λ ∈ (0, ∞), and one commonly considers the structure of G n, p in the limit as n → ∞. The original motivation for studying G n, p was to understand the properties of ‘typical’ graphs. This is in contrast to the study of ‘extremal’ graphs, although it may be noted that random graphs have on occasion manifested properties more extreme than graphs obtained by more constructive means. Random graphs have proved an important tool in the study of the ‘typical’ runtime of algorithms. Consider a computational problem associated with graphs, such as the travelling salesman problem. In assessing the speed of an algorithm for this problem, one may find that, in the worst situation, the algorithm is very slow. On the other hand, the typical runtime may be much less than the worst-case runtime. The measurement of ‘typical’ runtime requires a probability measure on the space of graphs, and it is in this regard that G n, p has risen to prominence within this subfield of theoretical computer science. While G n, p is, in a sense, the obvious candidate for such a probability measure, it suffers from the weakness 1 See

also [88].

c G. R. Grimmett 6 February 2009

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[11.2]

Random Graphs

that the ‘mother graph’ K n has a large automorphism group; it is a poor candidate in situations in which pairs of vertices may have differing relationships to one another. The random graph G n, p has received a very great deal of attention, largely within the community working on probabilistic combinatorics. The theory is based on a mix of combinatorial and probabilistic techniques, and has become very refined. One may think of G n, p as a percolation model on the complete graph K n . The parallel with percolation is weak in the sense that the theory of G n, p is largely combinatorial rather than geometrical. There is however a sense in which random graph theory has enriched percolation. The major difficulty in the study of physical systems arises out of the geometry of Rd ; points are related to one another in ways that depend greatly on their relative positions in R d . In a socalled ‘mean-field theory’, the geometrical component is removed through the assumption that points interact with all other points equally. Mean-field theory leads to a approximate picture of the model in question, and this approximation improves in the limit as d → ∞. The Erd˝os–R´enyi random graph may be seen as a mean-field approximation to percolation. Mean-field models based on G n, p have proved of value for Ising and Potts models also, see [41, 215]. The two principal references for the theory of G n, p are the earlier book [40] by Bollob´as, and the more recent work [129] of Janson, Łuzcak and Ruci n´ ski. We say nothing here about recent developments in random-graph theory involving models for the so-called small world. See [72] for example.

11.2 Giant component Consider the random graph G n,λ/n where λ ∈ (0, ∞) is a constant. We may build the component containing a given vertex v as follows. The vertex v is adjacent to a certain number N of vertices, where N has the bin(n−1, λ/n) distribution. Each of these vertices is joined to a random number of vertices, distributed approximately as N , and such that, with probability 1 − o(1), these new vertex-sets are disjoint. Since the bin(n − 1, λ/n) distribution is ‘nearly’ Poisson Po(λ), the component at v grows very much like a branching process with family-size distribution Po(λ). The branching-process approximation becomes less good as the component grows, and in particular when its size becomes of order n. The mean family-size equals λ, and thus the process with λ < 1 is very different from that with λ > 1. Suppose that λ < 1. In this case, the branching process is (almost surely) extinct, and possesses a finite number of vertices. Having built the component at v, one picks another vertex w and acts similarly. By iteration, one obtains that G n, p is the union of clusters each with exponentially decaying tail. The largest component has order log n. When λ > 1, the branching process grows beyond limits with strictly positive c G. R. Grimmett 6 February 2009

[11.2]

Giant component

181

probability. This corresponds to the existence in G n, p of a component of size having order n. We make this more formal as follows. Let X n be the number of vertices in a largest component of G n, p . We write Z n = op (yn ) if Z n /yn → 0 in probability as n → ∞. An event A n is said to occur asymptotically almost surely (abbreviated as a.a.s.) if P( A n ) → 1 as n → ∞. (11.1) Theorem [75]. We have that  op (1) 1 Xn = n α(λ)(1 + op (1))

if λ ≤ 1,

if λ > 1,

where α(λ) is the survival probability of a branching process with a single progenitor and family-size distribution Po(λ). It is standard (see [109, Sect. 5.4], for example) that the extinction probability η(λ) = 1 − α(λ) of such a branching process is the smallest non-negative root of the equation s = G(s) where G(s) = e λ(s−1) . It is left as an exercise2 to check that ∞ 1 X k k−1 η(λ) = (λe−λ )k . λ k! k=1

Proof. Since the distribution of X n is non-decreasing in λ, and since α(1) = 0, it suffices to consider the case λ > 1, and we assume this henceforth. We follow [129, Sect. 5.2], and use a branching-process argument. (See also [20].) Choose a vertex v. At the first step we find all neighbours of v, say v1 , v2 , . . . , vr , and we mark v as dead. At the second step, we generate all neighbours of v 1 in V \ {v, v1 , v2 , . . . , vr }, and we mark v1 as dead. This process is iterated until the entire component of G n, p containing v has been generated. Any vertex thus discovered in the component of v, but not yet dead, is said to be live. Step i is said to be complete when there are exactly i dead vertices. Conditional on the history of the process up to and including the (i − 1)th step, the number Ni of vertices added at step i is distributed as bin(n − m, p) where m is the number of vertices already generated. Let 16λ log n, k+ = n 2/3 . k− = (λ − 1)2 In this section, all logarithms are natural. Consider the above process started at v, and let Av be the event that: either the process terminates after fewer than k − 2 Here

is one way that resonates with random graphs. Let pk be the probability that vertex 1 lies in a component that is a tree of size k. By enumerating the possibilities, pk =

n − 1 k−2 k k−1


Simplify and sum over k. c G. R. Grimmett 6 February 2009



k λ k−1 λ k(n−k)+( 2)−k+1 1− . n n 





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Random Graphs

steps, or, for every k satisfying k − ≤ k ≤ k+ , there are at least 21 (λ − 1)k live vertices after step k. If Av does not occur, there exists k ∈ [k − , k+ ] such that: step k takes place and, after its completion, fewer than m = k + 21 (λ − 1)k = 21 (λ + 1)k vertices have been discovered in all. Thus, on A v , and with such a choice for k, (N1 , N2 , . . . , Nk ) ≥st (Y1 , Y2 , . . . , Yk ) where the Y j are independent random variables distributed as bin(n− 12 (λ+1)k, p). Therefore, k+ X 1 − P( Av ) ≤ πk , k=k−

where, by the Chernoff bound for the tail of the binomial distribution, for k − ≤ k ≤ k+ and large n, (11.2)

πk = P

X k i =1

Yi ≤

1 2 (λ



+ 1)k

(λ − 1)2 k 2 ≤ exp − 9λk = O(n −16/9 ).







(λ − 1)2 ≤ exp − k− 9λ



Therefore, 1 − P( Av ) ≤ k+ O(n −16/9 ) = o(n −1 ), and this proves that P

\

v∈V

Av



≥1−

X

v∈V

[1 − P( Av )] → 1

as n → ∞.

In particular, a.a.s., no component of G n,λ/n has size between k− and k+ . We show next that, a.a.s., T there do not exist more than two components with size exceeding k+ . Assume that v Av occurs, and let v 0 , v 00 be distinct vertices lying in components with size exceeding k + . We run the above process beginning at v 0 for the first k+ steps, and we finish with a set L 0 containing at least 21 (λ − 1)k+ live vertices. We do the same for the process from v 00 . Either the growing component at v 00 intersects the current component v 0 by step k+ , or not. If the latter, then, we finish with a set L 00 , containing at least 21 (λ − 1)k+ live vertices, and disjoint from L 0 . The chance (conditional on arriving at this stage) that there exists no edge between L 0 and L 00 is bounded above by
1 2 (1 − p)[ 2 (λ−1)k+ ] ≤ exp − 14 λ(λ − 1)2 n 1/3 = o(n −2 ).

c G. R. Grimmett 6 February 2009

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Giant component

183

Therefore, the probability that there exist two distinct vertices belonging to distinct components of size exceeding k + is no greater than \  1− P Av + n 2 o(n −2 ) = o(1). v∈V

In summary, a.a.s., every component is either ‘small’ (smaller than k − ) or ‘large’ (larger than k+ ), and there can be no more than one large component. In order to estimate the size of any such large component, we use Chebyshev’s inequality to estimate the aggregate sizes of small components. Let v ∈ V . The chance σ = σ (n, p) that v is in a small component satisfies (11.3)

η− − o(1) ≤ σ ≤ η+ ,

where η+ (respectively, η− ) is the extinction probability of a branching process with family-size distribution bin(n − k − , p) (respectively, bin(n, p)), and the o(1) term bounds the probability that the latter branching process terminates after k − or more steps. It is an easy exercise to show that η− , η+ → η as n → ∞ where η(λ) = 1 − α(λ) is the extinction probability of a Po(λ) branching process. The number S of vertices in small components satisfies E(S) = σ n = (1 + o(1))ηn.

Furthermore, by an argument similar to that above,   E(S(S − 1)) ≤ nσ k− + nσ (n − k− , p) = (1 + o(1))(E S)2 ,

whence, by Chebyshev’s inequality, G n, p possesses (η + op (1))n vertices in small components. This leaves just n − (η + op (1))n = (α + op (1))n vertices remaining for the large component, and the theorem is proved.  A further analysis yields the size X n of the largest subcritical component, and the size Yn of the second largest supercritical component. (11.4) Theorem. (a) When λ < 1, X n = (1 + op (1))

log n . λ − 1 − log λ

Yn = (1 + op (1))

log n − 1 − log λ0

(b) When λ > 1,

where λ0 = λ(1 − α(λ)).

λ0

If λ > 1, and we remove the largest component, we are left with a random graph on n − X n ∼ n(1 − α(λ)) vertices. The mean vertex-degree of this subgraph is approximately λ · n(1 − α(λ)) = λ(1 − α(λ)) = λ0 . n c G. R. Grimmett 6 February 2009

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It may be checked that this is strictly smaller than 1, implying that the remaining subgraph behaves as a subcritical random graph on n − X n vertices. Theorem 11.4(b) now follows from part (a). The picture is more interesting when λ ≈ 1, for which there is a detailed combinatorial study of [128]. Rather than describing this here, we deviate to the work of David Aldous [16], who has demonstrated a link, via the multiplicative coalescent, to Brownian motion. We set p=

1 t + 4/3 n n

where t ∈ R, and we write C nt (1) ≥ C nt (2) ≥ · · · for the component sizes of G n, p in decreasing order. We shall explore the weak limit (as n → ∞) of the sequence n −2/3 (C nt (1), C nt (2), . . . ). Let W = (W (s) : s ≥ 0) be a standard Brownian motion, and W t (s) = W (s) + ts − 21 s 2 ,

s ≥ 0,

a Brownian motion with drift t − s at time s. Write B t (s) = W t (s) − inf W t (s 0 ) 0≤s 0 ≤s

for a reflecting inhomogenous Brownian motion with drift. (11.5) Theorem [16]. As n → ∞, n −2/3 (C nt (1), C nt (2), . . . ) ⇒ (C t (1), C t (2), . . . ), where C t ( j ) is the length of the j th largest excursion of B t . We think of the sequences of Theorem 11.5 as being chosen at random from the space of decreasing non-negative sequences x = (x 1 , x 2 , . . . ), with metric d(x, y) =

sX i

(xi − yi )2 .

As t increases, two components of sizes x i , x j ‘coalesce’ at a rate proportional to the product xi x j . Theorem 11.5 identifies the scaling limit of this process as that of the evolving excursion-lengths of W t reflected at zero. This observation has contributed to the construction of the so-called ‘multiplicative coalescent’. In summary, the largest component of the subcritical random graph (when λ < 1) has order log n, and of the supercritical graph (when λ > 1) order n. When λ = 1, the largest component has order n 2/3 , with a multiplicative constant that is a random variable. The discontinuity at λ = 1 is sometimes referred to as the ‘Erd˝os–R´enyi double jump’. c G. R. Grimmett 6 February 2009

[11.3]

Independence and colouring

185

11.3 Independence and colouring Our second random-graph exercise concerns the chromatic number of G n, p for constant p. The chromatic number χ (G) of a graph G is the least number of colours with the property that: there exists an allocation of colours to vertices such that no two neighbours have the same colour. Let p ∈ (0, 1), and write χ n, p for the chromatic number of G n, p . A subset W of V is called independent if no pair of vertices in W are adjacent, that is, if X i, j = 0 for all i, j ∈ W . Any colouring of G n, p partitions V into independent sets, and therefore the chromatic number is related to the size I n, p of the largest independent set of G n, p . (11.6) Theorem [104]. We have that In, p = (1 + op (1))2 logπ n. where the base π of the logarithm is given by π = 1/(1 − p).

The proof follows a standard route: the upper bound follows by an estimate of an expectation, and the lower by an estimate of a second moment. When performed with greater care, such calculations yield much more accurate estimates of I n, p than those presented here, see, for example, [40], [129, Sect. 7.1], and [161, Sect. 2]. Specifically, there exists an integer-valued function r = r (n, p) such that

(11.7)

P(r − 1 ≤ In, p ≤ r ) → 1

as n → ∞.

Proof. Let Nk be the number of independent subsets of V with cardinality k. Then (11.8)

P(In, p ≥ k) = P(Nk ≥ 1) ≤ E(Nk ).

Now, (11.9)

  k n (1 − p)(2) , E(Nk ) = k

With  > 0, set k = 2(1 + ) logπ n, and use the fact that   nk n ≤ ≤ (ne/k)k , k k!

to obtain logπ E(Nk ) ≤ −(1 + o(1))k logπ n → −∞

as n → ∞.

By (11.8), P(In, p ≥ k) → 0 as n → ∞. This is an example of the use of the so-called ‘first-moment method’. c G. R. Grimmett 6 February 2009

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A lower bound for In, p is obtained by the ‘second-moment method’ as follows. By Chebyshev’s inequality,
var(Nk ) P(Nk = 0) ≤ P |Nk − E Nk | ≥ E Nk ≤ , E(Nk )2

whence, since Nk takes values in the non-negative integers, (11.10)

P(Nk ≥ 1) ≥ 2 −

E(Nk2 ) . E(Nk )2

Let  > 0 and k = 2(1 − ) logπ n. By (11.10), it suffices to show that E(Nk2 ) →1 E(Nk )2

(11.11)

as n → ∞.

By an elementary counting argument, E(Nk2 )

   k   k i k X k n−k n ( ) 2 (1 − p)(2)−(2) . (1 − p) = k −i i k i =0

After a minor analysis using (11.8) and (11.11), we deduce that P(I n, p ≥ k) → 1 as n → ∞. The theorem is proved.  We turn now to the chromatic number χn, p . Since the size of any set of vertices of given colour is no larger than In, p , one has trivially that (11.12)

χn, p ≥

n In, p

= (1 + op (1))

n . 2 logπ n

The sharpness of this inequality was proved by B´ela Bollob´as [39]. (11.13) Theorem [39]. We have that χn, p = (1 + op (1))

n . 2 logπ n

The term op (1) may be estimated quite precisely by a more detailed analysis than that presented here, see [39, 162] and [129, Sect. 7.3]. Specifically, one has, a.a.s., that n χn, p = , 2 logπ n − 2 logπ logπ n + Op (1) where Z n = Op (yn ) means P(|Z n /yn | > M) ≤ g(M) → 0 as M → ∞.

Proof. The lower bound follows as in (11.12), and so we concentrate on finding an upper bound for χn, p . Let 0 <  < 41 , and write k = b2(1 − ) logπ nc. We claim that, with probability 1 − o(1), every subset of V with cardinality at least c G. R. Grimmett 6 February 2009

[11.3]

Independence and colouring

187

m = bn/(logπ n)2 c possesses an independent subset of size at least k. The required bound on χn, p follows from this claim, as follows. We find an independent set of size k, and we colour its vertices with colour 1. From the remaining set of n − k vertices, we find an independent set of size k, and we colour it with colour 2. This process may be iterated until there remains a set S of size smaller than bn/(logπ n)2 c. We colour the vertices of S ‘greedily’, with |S| different colours. The total number of colours used is no larger than n n + , k (logπ n)2 which, for large n, is smaller than 21 (1 + 2)n/ logπ n. (11.14) Lemma. The probability that G m, p contains no independent set of size k
7 is less than exp −n 2 −2+o(1) /m 2 .
There are mn (< 2n ) subsets of {1, 2, . . . , n} with cardinality m. The probability that some such subset fails to contain an independent set of size k is, by the lemma, no larger than 2n exp(−n 2 −2+o(1) /m 2 ) = o(1). 7

We turn to the proof of Lemma 11.14, for which we shall use the Hoeffding inequality, Theorem 4.18. For M ≥ k, we write   k M F(M, k) = (1 − p)(2) . k We shall require M to be such that F(M, k) grows in the manner of a power of n, and to that end we set (11.15)

M = b(Ck/e)n 1− c,

where log C = has been chosen in such a way that (11.16)

3 log π 8(1 − ) 7

F(M, k) = n 4 −+o(1) .

Let I(r ) be the set of independent subsets of {1, 2, . . . , r } with size k. We write Nk = |I(m)|, and Nk0 for the number of elements I of I(m) with the property that |I ∩ I 0 | ≤ 1 for all I 0 ∈ I(m), I 0 6= I . Note that (11.17) c G. R. Grimmett 6 February 2009

Nk ≥ Nk0 .

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[11.3]

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We order as (e1 , e2 , . . . , e(m ) ) the edges of the complete graph on the vertex2 set {1, 2, . . . , m}. Let Fs be the σ -field generated by the states of the edges e1 , e2 , . . . , es , and let Ys = E(Nk0 | Fs ). It is elementary that the sequence m
(Ys , Fs ), 0 ≤ s ≤ 2 , is a martingale (see [109, Example 7.9.24]). The addition or removal of an edge may change N k0 by no more than 1, so the martingale differences satisfy |Ys+1 − Ys | ≤ 1. Since Y0 = E(Nk0 ) and Y(m ) = Nk0 , 2

(11.18)

P(Nk = 0) ≤

P(Nk0 P Nk0

= 0)

− E(Nk0 ) ≤ −E(Nk0 )     m 0 2 1 ≤ exp − 2 E(Nk ) 2
0 2 2 ≤ exp −2E(Nk ) /m , =




by (11.17) and Theorem 4.18. We now require a lower bound for E(N k0 ). Let M be as in (11.15). Let Mk = |I(M)|, and let Mk0 be the number of elements I ∈ I(M) such that |I ∩ I 0 | ≤ 1 for all I 0 ∈ I(M), I 0 6= I . Clearly (11.19)

Nk0 ≥ Mk0 ,

and we shall bound E(Mk0 ) below. Let K = {1, 2, . . . , k}, and let A be the event that K is an independent set. Let Z be the number of elements of I(M), other than K , that intersect K in two or more vertices. Then   M 0 (11.20) P( A ∩ {Z = 0}) E(Mk ) = k   M = P( A)P(Z = 0 | A) k = F(M, k)P(Z = 0 | A). We bound P(Z = 0 | A) by

(11.21)

P(Z = 0 | A) = 1 − P(Z ≥ 1 | A)

≥ 1 − E(Z | A)  k−1   X k t k M −k (1 − p)(2)−(2) =1− k−t t t=2

=1− For t ≥ 2,

k−1 X

Ft ,

say.

t=2

  1 (M − 2k + 2)! (k − 2)! 2 2 (11.22) Ft /F2 = · · (1 − p)− 2 (t+1)(t−2) (M − 2k + t)! (k − t)! t! " # t−2 1 k 2 (1 − p)− 2 (t+1) ≤ . M − 2k c G. R. Grimmett 6 February 2009

[11.3]

Independence and colouring

189

For 2 ≤ t ≤ 21 k,

  1 logπ (1 − p)− 2 (t+1) ≤ 41 (k + 1) <

1 2

+ 21 (1 − ) logπ n,

so (1 − p)− 2 (t+1) = o(n 2 ). By (11.22), X Ft = (1 + o(1))F2 . 1

1

2≤t≤ 21 k

Similarly, Ft /Fk−1

   1 k M − k (1 − p) 2 (k+t−2)(k−t−1) = k(M − k) t k −t   1 k−t−1 ≤ kn(1 − p) 2 (k+t−2) .

For 21 k ≤ t ≤ k − 1, we have as above that 1

3

9

(1 − p) 2 (k+t) ≤ (1 − p) 4 k ≤ n − 8 whence

X

1 k
In summary, (11.23)

k−1 X t=2

Ft = (1 + o(1))Fk−1 .

Ft = (1 + o(1))(F2 + Fk−1 ).

By (11.16), F2 ≤

k4 F(M, k) 2(1 − p)(M − k)2

= n − 4 ++o(1) = o(1), 1

and similarly

Fk−1 = k(M − k)(1 − p)k−1 = o(1).

By (11.23) and (11.20)–(11.21),

E(Mk0 ) = (1 + o(1))F(M, k) = n 4 −+o(1) . 7

Returning to the martingale bound (11.18), it follows by (11.19) that P(Nk = 0) ≤ exp(−n 2 −2+o(1) /m 2 ) 7

as required. c G. R. Grimmett 6 February 2009



190

[11.4]

Random Graphs

11.4 Exercises 11.1. Let η(λ) be the extinction probability of a branching process whose family-sizes are distributed as Po(λ). Show that ∞

1 X k k−1 η(λ) = (λe−λ )k . λ k! k=1

11.2. [129] Chernoff bounds. Let X be distributed as bin(n, p) and λ = np. Show that P(X ≥ E(X ) + t) ≤ e

−λφ(t/λ)

P(X ≤ E(X ) − t) ≤ e−λφ(−t/λ) where φ(x) =





 t2 ≤ exp − , 2(λ + t/3)  2 t ≤ exp − , λ

(1 + x) log(1 + x) − x ∞

t ≥ 0, t ≥ 0,

if x ≥ −1, if x < −1.

11.3. Consider a branching process with family-size distribution bin(n, λ/n). Show that the extinction probability converges to η(λ) as n → ∞, where η(λ) is the extinction probability of a Po(λ) branching process. 11.4. [40] Show that the size of the largest independent set of G n, p is, a.a.s., either r − 1 or r , for some deterministic function r = r (n, p).

c G. R. Grimmett 6 February 2009

12 Lorentz Gas

A small particle is fired through an environment of large particles, and is subjected to reflections on impact. Little is known about the trajectory of the small particle when the larger ones are distributed at random. The notorious problem on the square lattice is summarized, and open questions are posed for the case of a continuum of needle-like mirrors in the plane.

12.1 Lorentz model In a famous sequence [154] of papers of 1906, Hendrik Lorentz introduced the following problem. Large (heavy) particles are distributed about R d . A small (light) particle is fired through Rd , with a trajectory comprising straight-line segments between the points of interaction with the heavy particles. When the small particle hits a heavy particle, the small particle is reflected, and the large particle remains motionless. See Figure 12.1 for an illustration. We may think of the heavy particles as objects bounded by reflecting surfaces, and the light particle as a photon. The problem is to say something non-trivial about how the trajectory of the photon depends on the ‘environment’ of heavy particles. Conditional on the enviroment, the photon pursues a deterministic path about which the natural questions include: 1. is the path unbounded? 2. how distant is the photon from its starting point after the elapse of time t? For simplicity, we assume henceforth that the large particles are identical to one another, and the small particle is of negligible volume. Probability may be injected naturally into this model through the assumption that the heavy particles are distributed at random around Rd according to some probability measure µ. The questions above may be rephrased, and made more precise, in the language of probability theory. Let X t denote the position of the photon at time t, assuming constant velocity. Under what conditions on µ: I. is there strictly positive probability that the function X t is unbounded? c G. R. Grimmett 6 February 2009

192

[12.2]

Lorentz Gas

Figure 12.1. The trajectory of the photon comprises straight-line segments between the points of reflection.

II. does X t converge to a Brownian motion, after suitable re-scaling? For a wide choice of measures µ, these questions are currently unanswered. The Lorentz gas is very challenging to mathematicians, and little is known rigorously in answer to the questions above. The reason is that, as the photon moves around space, it gathers information about the random environment, and it carries this information with it for ever more. The Lorentz gas was developed by Paul Ehrenfest [74]. For the relevant references in the mathematics and physics journals, the reader is referred to [95, 96]. Many references may be found in [202].

12.2 The square Lorentz gas Probably the most provocative version of the Lorentz gas for probabilists arises when the light ray is confined to the square lattice L2 . At each vertex v of L2 , we place a ‘reflector’ with probability p, and nothing otherwise (the occupancies of different vertices are independent). Reflectors come in two types: ‘NW’ and ‘NE’. A NW reflector deflects incoming rays heading northwards (respectively, southwards) to the west (respectively, east) and vice versa. NE reflectors behave similarly with east and west interchanged. See Figure 12.2. Think of a reflector as being a two-sided mirror placed at 45◦ to the axes, so that an incoming light ray is reflected along an axis perpendicular to its direction of arrival. Now, for each vertex x, with probability p we place a reflector at x, and otherwise we place nothing at x. This is done independently for different x. If a reflector is placed at x, then we specify that it is equally likely to be NW as NE. We shine a torch northwards from the origin. The light is reflected by the mirrors, and we ask whether or not the light ray returns to the origin. Letting η( p) = Pp (the light ray returns to the origin), we ask when is it the the case that η( p) = 1. It is reasonable to conjecture that η c G. R. Grimmett 6 February 2009

[12.2]

The square Lorentz gas

NW

193

NE

Figure 12.2. An illustration of the effects of NW and NE reflectors on the light ray.

is non-decreasing in p. Certainly η(0) = 0, and it is well known 1 that η(1) = 1. (12.1) Theorem. We have that η(1) = 1.

We invite the reader to consider whether or not η( p) = 1 for some p ∈ (0, 1). A variety of related conjectures, not entirely self-consistent, may be found in the physics literature. There are almost no mathematical results about this process beyond Theorem 12.1. We mention the paper [179], where it is proved that the number N ( p) of unbounded light rays on Z2 is almost surely constant, and is equal to one of 0, 1, ∞. Furthermore, if there exist unbounded light trajectories, then they self-intersect infinitely often. If N ( p) = ∞, the position X n of the photon at time n, when following an unbounded trajectory, is superdiffusive in the sense that E(|X n |2 )/n is unbounded as n → ∞. The principal method of [179] is to observe the environment of mirrors as viewed from the moving photon. In a variant of the standard random walk, termed the ‘burn-your-bridges’ random walk by Omer Angel, an edge is destroyed immediately after it is traversed for the first time. When p = 31 , the photon follows a burn-your-bridges random walk on L2 . Proof. We construct an ancillary lattice L as follows. Let  A = (m + 21 , n + 21 ) : m + n is even .

Let ∼ be the adjacency relation on A given by(m + 12 , n + 12 ) ∼ (r + 12 , s + 21 ) if and only if |m − r | = |n − s| = 1. We obtain thus a graph L on A that is isomorphic to L2 . See Figure 12.3. We declare the edge of L joining (m − 21 , n − 21 ) to (m + 21 , n + 12 ) to be open if there is a NE mirror at (m, n); similarly we declare the edge joining (m − 21 , n + 12 ) to (m + 21 , n − 12 ) to be open if there is a NW mirror at (m, n). Edges that are not open are designated closed. This defines a bond percolation process in which north-easterly and north-westerly edges are open with probability 1 1 2 2 . Since pc (L ) = 2 , the process is critical. See Section 5.5. 1 See

the historical remark in [94].

c G. R. Grimmett 6 February 2009

194

[12.3]

Lorentz Gas

0

0

Figure 12.3. (a) The heavy lines form the lattice L, and the central point is the origin of 2 . (b) An open circuit in L constitutes a barrier of mirrors through which no light may penetrate.

Let N be the number of open circuits in L with the origin in their interiors. By the remarks prior to Theorem 5.33, we have that P(N ≥ 1) = 1, where P is the appropriate probability measure. Such an open circuit corresponds to a barrier of mirrors surrounding the origin (see Figure 12.3), from which no light can escape. Therefore η(1) = 1.  The problem above may be stated for other lattices such as Ld , see [94] for example. It is much simplified if one allows the photon to flip its own coin as it proceeds through the disordered medium of mirrors. Two such models have been explored. In the first, there is a positive probability that the photon will misbehave when it hits a mirror, see [207]. In the second, there is allowed a small density of vertices at which the photon acts in the manner of a random walk, see [35, 105].

12.3 In the plane Here is a continuum version of the Lorentz gas. Let 5 be a Poisson process in R 2 with intensity 1. For each x ∈ 5, we place a needle (that is, a closed rectilinear line-segment) of given length l with its centre at x. The orientations of the needles are taken to be independent random variables with a common law µ on [0, π ). We call µ degenerate if it has support on a singleton, that is, all needles are (almost surely) parallel. Each needle is interpreted as a (two-sided) reflector of light. Needles are permitted to overlap. Light is projected from the origin northwards, and deflected by the needles. Since the light strikes the endpoint of some needle with probability 0, we shall overlook this possibility. In a related problem, we may study the union M of the needles, viewed as subsets of R2 , and ask whether either (or both) of the sets M, R2 \ M contains an unbounded component. This problem is known as ‘needle percolation’, and it has received some attention (see, for example, [163, Sect. 8.5], and also [120]). Of c G. R. Grimmett 6 February 2009

[12.3]

In the plane

195

concern to us in the present setting is the following. Let λ(l) be the probability that there exists an unbounded path of R2 \ M with the origin 0 as endpoint. It is clear that λ(l) is non-increasing in l. The following is a fairly straightforward exercise of percolation type. (12.2) Theorem [120]. There exists l c = lc (µ) ∈ (0, ∞] such that  > 0 if l < lc , λ(l) = 0 if l > lc , and furthermore lc < ∞ if and only if µ is non-degenerate.

The phase transition has been defined here in terms of the existence of an unbounded ‘vacant path’ from the origin. When no such path exists, the origin is almost surely surrounded by a cycle of pairwise-intersecting needles. That is,  < 1 if l < lc , (12.3) Pµ (E) = 1 if l > lc ,

where E is the event that there exists a component C of needles such that the origin of R2 lies in a bounded component of R2 \ C, and Pµ denotes the probability measure governing the configuration of mirrors. The needle percolation problem is a type of continuum percolation model, cf. the space–time percolation process of Section 6.6. Continuum percolation, and in particular the needle (or ‘stick’) model, has been summarized in [163, Sect. 8.5]. We return to the above Lorentz problem. Suppose that the photon is projected from the origin at angle θ to the x-axis, for given θ ∈ [0, 2π ). Let 2 be the set of all θ such that the trajectory of the photon is unbounded. It is clear from Theorem 12.2 that Pµ (2 = ∅) = 1 if l > lc . The strength of the following theorem of Matthew Harris lies in the converse statement. (12.4) Theorem [120]. Let µ be non-degenerate, with support a subset of the rational angles π Q. (a) If l > lc , then Pµ (2 = ∅) = 1. (b) If l < lc , then P(2 has Lebesgue measure 2π ) = 1 − Pµ (E) > 0.

That is to say, almost surely on the complement of E, the set 2 differs from the entire interval [0, 2π ) by a null set. The proof uses a type of dimensionreduction method, and utilizes a theorem concerning so-called ‘interval-exchange transformations’ taken from ergodic theory, see [134]. It is a key assumption for this argument that µ be supported within the rational angles. Theorem 12.4 leaves open even the arguably most natural instance of the problem, in which µ is uniform on [0, π ). Let η(l) be the probability that the light ray is bounded, having started by heading northwards from the origin. As above, c G. R. Grimmett 6 February 2009

196

[12.4]

Lorentz Gas

η(l) = 1 when l > lc . In contrast, it is not known for general µ whether or not η(l) < 1 for sufficiently small positive l. It seems reasonable to conjecture the following. For any probability measure µ on [0, π ), there exists l r ∈ (0, lc ] such that η(l) < 1 whenever l < l r . This conjecture is open even for the case when µ is uniform on [0, π ). (12.5) Conjecture. Let µ be the uniform probability measure on [0, π ), and let lc denote the critical length for the associated needle percolation problem (as in Theorem 12.2). (a) There exists lr ∈ (0, lc ] such that η(l)



<1

if l < lr ,

=1

if l > lr ,

(b) We have that lr = lc .

As a first step, we seek a proof that η(l) < 1 for sufficiently small positive values of l. It is typical of such mirror problems that we lack even a proof that η(l) is monotone in l.

12.4 Exercises 12.1. There are two ways of putting in the barriers in the percolation proof of Theorem 12.1, depending on whether one uses the odd or the even vertices. Use this fact to establish bounds for the tail of the size of the trajectory when the density of mirrors is 1. 12.2. In a variant of the square Lorentz lattice gas, NE mirrors occur with probability η and NW mirrors otherwise. Show that the photon’s trajectory is almost surely bounded. 12.3. Needles are dropped in the plane in the manner of a Poisson process with intensity 1. They have length l, and their angles to the horizontal are independent random variables with law µ. Show that there exists l c = lc (µ) ∈ (0, ∞] such that: the probability that the origin lies in an unbounded path intersecting no needle is strictly positive when l < l c , and equals zero when l > l c . 12.4. (continuation) Show that l c < ∞ if and only if µ is non-degenerate.

c G. R. Grimmett 6 February 2009

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c G. R. Grimmett 6 February 2009

Index

Aldous–Broder algorithm 21, 31 amenability 25 asymptotically almost surely (a.a.s.) 183 BK inequality 48 block lattice 110 Boolean function 56 Borel σ -algebra 26 boundary conditions 47 bounded differences 50 branching process 182 bridge 125 Brownian Motion 27, 29, 186 burn-your-bridges random walk 195 Cardy formula 29, 99, 100 Cauchy–Riemann equations 105 Chernoff bound 184 chromatic number 71, 187 clique 128 closed edge 33 cluster 33 cluster-weighting factor 159 comparison inequalities 139 configuration 33 maximum, minimum c. 45 partial order 44 conformal field theory 29 connective constant 37 contact model 114, 172 c. m. on tree 121 correlation–connection theorem 138 coupling additive c. 117 c. of percolation 39

c. of contact models 115 c. of random-cluster and Potts models 137 monotone c. 117 critical exponent 86 critical point 141 critical probability 34 for random-cluster measures 141 in two dimensions 93, 108 on slab 78 Curie point 133 cut 125 cylinder event 26, 32 density matrix 158 reduced d. m. 166 dependency graph 128 disjoint-occurrence property 48 DLR-random-cluster-measure 140 dual graph 35 duality d. for contact model 117, 172 d. for voter model 173 effective conductance/resistance 8 convexity of e. r. 9, 16 electrical network 1, 3 parallel/series laws 8 energy 7 entanglement 167 epidemic 114 Erd˝os–R´enyi random graph 181 ergodic measure 24, 140, 155, 174 even graph 151 event

Index

A-invariant e. 69 exclusion model 176 exploration process 105 exponential decay of radius 73, 127 extremal measure 117, 174 ferromagnetism 133 flow 5 finite-energy property 83 first-order phase transition 142 first-passage percolation 52 FK representation 137 FKG inequality 47, 139 lattice condition/property 47, 139 flow 5 Fourier–Walsh coefficients 56 giant component 182 Gibbs, J. W. 130 Gibbs state 130, 133 for Ising model 134 for Potts model 134 graphical representation 116, 173 ground state 158, 165 Hamiltonian 134, 158 Hammersley–Simon–Lieb inequality 74 harmonic function 2, 108 Heisenberg model 134 hexagonal lattice 37, 90 Hille–Yosida Theorem 171 Hoeffding inequality 50, 189 Holley inequality 45 hull 28 hypercontractivity 58 hyperscaling relations 87 increasing event, random variable 19, 44, 159 independent set 187 infinite-volume limit 139 influence i. of edge 51 i. theorem 52 invariant measure 117, 171 Ising model 134, 133 continuum I. m. 160

quantum I. m. 157 spin cluster 148, 149 Kirchhoff laws 3 Theorem 5 lace expansion 78 large deviations Chernoff bound 184 logarithmic asymptotics 86 loop -erased random walk 20, 30 -erasure 20 Lorentz gas 193 magnetization 143 Markov chain 1, 171 generator 171 reversible M. c. 1 Markov field 130 Markov property 130 maximum principle 16 mean-field theory 182, 88 mixing measure 154 multiplicative coalescent 186 multiscale analysis 127 n-vector model 134 needle percolation 197 negative association 18, 19 O(n) model 134 Ohm’s law 4 open cluster 33 edge 33 oriented percolation 39 parallel/series laws 8 partial order 134 Peierls argument 36 percolation model 33 of words 43 p. probability 34, 39, 118, 141, 166 phase transition 86 first/second order p. t. 142

211

212

Index

photon 193 pivotal element/edge 68 planar dual 35 P´olya Theorem 13 portmanteau theorem 26 positive association 47 potential function 4, 130 Potts model 134, 133 continuum P. m. 160 ergodicity and mixing 155 magnetization 143 two-point correlation function 138 probability measure A-invariant p. m. 69 product σ -algebra 26 quantum Ising model 157 radius 110, 125 exponential decay of r. 73 random even graph 151 random graph 181 chromatic number 187 double jump 186 giant component 182 independent set 187 random-cluster measure 136 comparison inequalities 139 conditional measures 139 critical point 141 DLR measure 140 ergodicity 140, 155 limit measure 140 mixing 154 ratio weak-mixing 168 relationship to Potts model 138 uniqueness 141 random-cluster model 135 continuum r.-c. m. 158 r.-c. m. in two dimensions 146 random-current representation 153 random walk 1, 174 ratio weak-mixing 168 Rayleigh principle 9, 19, 24 Reimer inequality 49 renormalized lattice 110 root 21 roughening transition 145

RSW theorem 91 Russo formula 68 scaling relations 87 Schramm–L¨owner evolution, see SLE second-order phase transition 142 self-avoiding walk (SAW) 34, 37 self-dual graph 36 point 92, 147, 160 semigroup 171 series/parallel laws 8 sharp threshold theorem 69 sink 3, 5 slab critical point 78 SLE 27, 86 chordal, radial 28 source 3, 5 space–time percolation 124, 158 spanning s. arborescence 21 s. tree 5 square lattice 13 star–triangle transformation 9, 16 stochastic L¨owner evolution, see SLE stochastic ordering 44 Strassen theorem 45 strong survival 122 subadditive inequality 37 subcritical phase 73 supercritical phase 78 superposition principle 4 Thomson principle 9 time-constant 53 trace 158 transitive action 69 triangle condition 89 triangular lattice 29, 90 trifurcation 84 uniform connected subgraph (UCS) 26 spanning tree (UST) 18 (spanning) forest (USF) 26 uniqueness u. of infinite open cluster 83 u. of random-cluster measure 141

Index

universality 88 upper critical dimension 87 Voronoi percolation 53 voter model 173 weak convergence 25

c G. R. Grimmett 6 February 2009

weak survival 122 Wilson algorithm 20 word percolation 43 zero/one-infinite-cluster property 140

213